Paper Title: Bounding adapted Wasserstein metrics.













 
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Bounding adapted Wasserstein metrics
    Jose Blanchet, Martin Larsson, Jonghwa Park, Johannes Wiesel
    August 5, 2024
================================================================



§ ABSTRACT 
The Wasserstein distance 𝒲_p is an important instance of an optimal transport cost. Its numerous mathematical properties as well as applications to various fields such as mathematical finance and statistics have been well studied in recent years. The adapted Wasserstein distance 𝒜𝒲_p extends this theory to laws of discrete time stochastic processes in their natural filtrations, making it particularly well suited for analyzing time-dependent stochastic optimization problems.
    
    While the topological differences between 𝒜𝒲_p and 𝒲_p are well understood, their differences as metrics remain largely unexplored beyond the trivial bound 𝒲_p≲𝒜𝒲_p. This paper closes this gap by providing upper bounds of 𝒜𝒲_p in terms of 𝒲_p through investigation of the smooth adapted Wasserstein distance. Our upper bounds are explicit and are given by a sum of 𝒲_p, Eder's modulus of continuity and a term characterizing the tail behavior of measures. As a consequence, upper bounds on 𝒲_p automatically hold for 𝒜𝒲_p under mild regularity assumptions on the measures considered. A particular instance of our findings is the inequality 𝒜𝒲_1≤ C√(𝒲_1) on the set of measures that have Lipschitz kernels. 

    Our work also reveals how smoothing of measures affects the adapted weak topology. In fact, we find that the topology induced by the smooth adapted Wasserstein distance exhibits a non-trivial interpolation property, which we characterize explicitly: it lies in between the adapted weak topology and the weak topology, and the inclusion is governed by the decay of the smoothing parameter.

    
    

    Keywords: (adapted) Wassestein distance, optimal transport, modulus of continuity, relative compactness




Introduction

For N∈ we denote by _p(^N) the set of all Borel probability measures on ^N that have finite p-moments, where 1≤ p<∞ is fixed throughout this paper. The Wasserstein distance, defined via 

    _p(μ, ν)
        :=(inf_π∈(μ, ν)∫_^N×^Nx-y^p π(dx, dy))^1/p,

is a metric on _p(^N) that metrizes weak convergence plus convergence of p-moments. Here · denotes the Euclidean norm on ^N and (μ, ν) is the set of all couplings between μ∈_p(^N) and ν∈_p(^N), i.e., those probability measures on ^N×^N whose first marginal is μ and whose second marginal is ν. Computing the Wasserstein distance is an instance of the Kantorovich optimal transport problem <cit.>, and its numerous mathematical properties are well studied. We refer to <cit.> for a general overview.

While the Wasserstein distance between laws on ^N has seen a surge of applications in the last few years, the situation changes  when μ,ν are interpreted as laws of discrete-time stochastic processes in their natural filtrations. In this case, measuring the distance between μ and ν via _p(μ, ν) is often inadequate when considering time-dependent optimization problems such as optimal stopping problems and multistage optimization. The key reason for this is that the formulation (<ref>) does not take the time-structure of the laws μ,ν into account when optimizing over all possible couplings on ^N. The restriction to so-called bicausal or adapted couplings described in the next paragraph addresses this issue.

Let us first  set up some notation: consider two stochastic processes X=(X_1, …, X_T) and Y=(Y_1,…, Y_T) in finite discrete time, where both X_t and Y_t are ^d-valued random variables for t∈{1,2,…, T}. Throughout the paper, T∈ denotes the number of time steps and d∈ is the dimension of the state space. Consider now a Monge map S=(S_1, …, S_T):(^d)^T→ (^d)^T from the law of X to the law of Y, which satisfies

    Y_td= S_t(X_1, …, X_T)    for  t∈{1,2,…, T}.

Without further restrictions on the map S, Y_t can depend on the whole stochastic process X. In other words, it is impossible to  determine Y_t only from the information (X_1, …, X_t) as the map S_t uses information about the evolution of X after time t in general. In order to respect the filtration generated by X, it is reasonable to require S to be adapted, i.e., S_t should only be a function of the first t coordinates. As shown e.g. in <cit.>, the notion of adapted mappings is generalized to bicausal couplings as follows.


Let μ and ν be two probability measures on (^d)^T. A coupling π∈(μ, ν) is called bicausal if for (X,Y)∼π and t∈{1,2,…, T-1},

    (Y_1, …, Y_t)  and  (X_t+1, …, X_T)  are conditionally independent given  X_1, …, X_t

and

    (X_1, …, X_t)  and  (Y_t+1, …, Y_T)  are conditionally independent given  Y_1, …, Y_t.

The set of all bicausal couplings between μ and ν is denoted by (μ, ν). 


Heuristically speaking, π∈(μ, ν) couples two processes X∼μ and Y∼ν such that predicting Y at time t requires at most the information of X up to time t and vice versa. We define the adapted Wasserstein distance as the optimal transport cost over bicausal couplings.
[The adapted Wassertein distance]
    Let μ, ν∈_p((^d)^T). The adapted Wasserstein distance between μ and ν is given by
    
    _p(μ, ν)
            :=(inf_π∈(μ, ν)∫_(^d)^T× (^d)^T∑_t=1^Tx_t-y_t^p π(dx, dy))^1/p.



Much of the previous work has been on characterizing the topology generated by _p; we provide a literature review of these results in Section <ref>. Here we only remark that by Definition <ref> one can easily derive the inequality _p≤ C_p (for some constant C>0 depending only on p, T). As a consequence, the adapted Wasserstein topology is finer than the Wasserstein topology. The following example going back at least to <cit.> shows that this inclusion is strict in general. We refer to it as the standard example and use it to illustrate our results throughout this paper.

[Standard example]
    Let μ_=1/2δ_(, 1)+1/2δ_(-, -1) and μ=1/2δ_(0,1)+1/2δ_(0,-1). As → 0 we have _p(μ, μ_)=→ 0, but _p(μ, μ_)=(^p+2^p-1)^1/p↛ 0. 






In fact, the adapted Wasserstein topology is the coarsest topology that makes optimal stopping problems continuous <cit.>. <cit.> demonstrate, that time-dependent optimization problems from mathematical finance such as utility maximization, pricing and hedging derivatives, risk measurements, etc., are actually Lipschitz continuous with respect to _p.

While the differences between the Wasserstein topology and the adapted Wasserstein topology are thus well understood, the comparison between the metrics _p and _p is still largely open to the best of our knowledge. This article aims to fill this gap. We want to answer the following question:


Can we identify a class of subsets K⊆_p((^d)^T), on which _p and _p are nearly equivalent, i.e. there are “nice" explicit upper bounds of _p in terms of _p?


Except for trivial bounds, the above is an ill-posed task for K=_p((^d)^T). In fact, Example <ref> clearly shows that _p and _p cannot be equivalent in any meaningful sense. However, this example is arguably tailor made to showcase the differences between _p and _p. In fact, {μ_:>0} are chosen in such a way that the conditional distributions x_1↦(X^_2∈ · |X^_1=x_1) become increasingly irregular for (X^_1, X^_2)∼μ_: recall that we have

    (X^_2= 1|X^_1=ϵ)=1  and  (X^_2=-1|X^_1=-ϵ)=1.

In other words, {μ_: >0} does not have equicontinuous kernels. In many situations it is however reasonable to restrict to subsets  K⊆_p((^d)^T), whose elements have reasonably smooth kernels. It turns out that such subsets K are exactly the _p-relatively compact sets.



A set K⊆_p((^d)^T) is _p-relatively compact if and only if K is _p-relatively compact and

    lim_δ→ 0sup_μ∈ Kω^t,p_μ(δ)=0
     for all  t∈{1,2,…, T-1},

where ω^t,p_μ:(0,∞)→ (0,∞) is the p-modulus of continuity of μ defined via

    ω_μ^t,p(δ)
                :=sup{([_p((X_t+1∈ ·  X_1, …, X_t), (Y_t+1∈ ·  Y_1, …, Y_t))^p])^1/p}.

Here, the supremum is taken over all X,Y∼μ such that (X_1, …, X_t)-(Y_1, …, Y_t)_L^p<δ.


Let us emphasize here that focusing on the _p-relatively compact subsets of _p((^d)^T) is usually not a big restriction in practical applications, and sometimes even arises naturally. We refer to Section <ref> for a more detailed discussion.

Proposition <ref> will prove central in this article, as we will essentially show that the inequality _p≤ C_p can be reversed exactly on the _p-relatively compact sets. To get some intuition for this, assume for now that p=1 and K⊆_1((^d)^T) is _1-compact. Then it essentially follows from <cit.>, that the Wasserstein topology and the adapted Wasserstein topology agree on K. As a consequence, for any σ>0 there exists a constant C depending on σ and K such that we have

    _1(μ,ν)≤ C_1(μ,ν) +σ

for all μ, ν∈ K. While theoretically appealing, this result is of very limited practical relevance as long as one cannot characterize the dependence of C on σ and K. One of the main contributions of this article is to make the bound (<ref>) explicit. In fact, we show that (ignoring dimensional constants that can be made explicit) for any positive real numbers 0<σ<R, 

    _1(μ,ν)
    ≲R/σ_1(μ,ν) + h(σ,K) + g(R,K),

where the function h is derived from Eder's modulus of continuity ω (see Proposition <ref>) and the function g characterizes the tail behavior of measures in K outside a Euclidean ball of radius R. In particular, lim_σ→ 0 h(σ,K)=0 and lim_R→∞ g(R,K) =0. We refer to Corollary <ref> for an exact statement of this bound.

In other words, (<ref>) implies that _1 can be bounded by _1 up to an error that is uniformly bounded on _1-relatively compact sets. This seemingly simple result shows its true power in applications: the time-dependent optimization problems (e.g. utility maximization, pricing, hedging) mentioned above are not just continuous in _1, but are actually continuous in _1 on _1-relatively compact sets and estimates are explicitly computable. This significantly simplifies the analysis of time-dependent optimization problems. Indeed, it is actually not necessary to design new estimates specifically tailored for _1 as e.g. done in <cit.>; instead one can simply rely on the rich theory that has already been developed for _1 in the last decades. We give some examples illustrating this new methodology below, showing how one can derive finite-sample guarantees for the adapted Wasserstein distance. We believe that  (<ref>) has much further-reaching implications for the theory of adapted transports however, which we aim to investigate in future work.



 §.§ The smooth adapted Wasserstein distance


In order to derive (<ref>), we first need to introduce a variant  of _p: 


Given a noise distribution ξ and a bandwidth parameter σ>0, we define the smooth adapted Wasserstein distance via 
    ^(σ)_p(μ, ν):=_p(μ∗ξ_σ, ν∗ξ_σ).

Here ∗ denotes the convolution of two measures and ξ_σ is the law of (x↦σ x)_#ξ, i.e. if Z∼ξ, then σ Z∼ξ_σ.


 The non-adapted counterpart of ^(σ)_p is called the smooth Wasserstein distance 
    ^(σ)_p(μ, ν):=_p(μ∗ξ_σ, ν∗ξ_σ)
 and is by now well studied in the statistics and machine learning literature, mainly for the purpose of relaxing the curse of dimensionality. Indeed, denoting the empirical measure of μ by μ_n:=1/n∑_j=1^nδ_X^(j) where X^(1), X^(2), … are i.i.d samples of μ, it turns out that √(n)^(σ)_p(μ, μ_n) has a weak limit under suitable moment assumptions. We refer to <cit.> and the references therein for this line of research. On the other hand, ^(σ)_p is quite new to the best of our knowledge, and has only been studied very recently in the literature, cf. <cit.>. These results suggest that, as in the non-adapted case, the empirical convergence rate only exhibits a mild dependence on the dimension after smoothing.

While the statistical properties of ^(σ)_p(μ_n, μ) are thus at least partially understood, the arguably more fundamental question of how smoothing affects the adapted Wasserstein topology is still wide open to the best of our knowledge. Recall that it is well known that _p and ^(σ)_p generate the same topology (see <cit.>), and it is thus natural to conjecture that the same holds for _p and ^(σ)_p. Perhaps rather surprisingly, we show that this conjecture is false in general. In fact we prove that, essentially,

    Wasserstein topology ⊆ smooth adapted Wasserstein topology ⊆ adapted Wasserstein topology,

and the inclusions in the above display are equalities under suitable assumptions on the decay of σ (when thinking about sequential convergence). We return to this discussion in Section <ref>.



 §.§ TEXT
 

We are now in a position to give  a heuristic derivation of (<ref>). To keep the notation simple, we take p=1, T=2 and refer to Corollary <ref> for the general statement. Assuming K⊆_1((^d)^2) is _1-relatively compact, an application of the triangle inequality yields

    _1(μ, ν)
        ≤^(σ)_1(μ, ν)
        +2sup_μ∈ K_1(μ, μ∗ξ_σ)

for any μ, ν∈ K and σ>0. In order to bound _1, it is thus sufficient to bound the two terms on the right-hand side of (<ref>) separately. These bounds constitute the two main results of our paper and we refer the reader to Section <ref> for exact statements. We start with a bound on the smooth adapted Wasserstein distance.

 
For any σ, R>0 we have

    ^(σ)_1(μ, ν)
        ≲R/σ_1(μ, ν)
        +sup_μ∈ K∫_{x≥ R}x (μ∗ξ_σ)(dx).


In other words, ^(σ)_1 can be bounded by _1 up to an error term depending only on the tails of μ∗ξ_σ. In the next section we show that the scaling of order _1/σ in the upper bound is essentially optimal. Our second main result pertains to the question, how much information is lost by considering μ∗ξ_σ instead of its unperturbed version μ. 


We have

    sup_μ∈ K_1(μ, μ∗ξ_σ)
        ≲σ+sup_μ∈ Kω^1,1_μ(σ).



In conclusion, the difference between μ and its smoothed version μ∗ξ_σ is bounded by Eder's modulus of continuity ω^1,1_μ uniformly on the _1-relatively compact sets.

Combining Theorems <ref> and <ref> with  (<ref>) finally yields the following result:


For all 0<σ<R we have

    _1(μ, ν)
        ≲R/σ_1(μ, ν)
        +(σ+sup_μ∈ Kω^1,1_μ(σ))
        +sup_μ∈ K∫_{x≥ R}xμ(dx).



Returning to (<ref>), we conclude that we can choose

    h(σ, K)
        =σ+sup_μ∈ Kω^1,1_μ(σ),  
        g(R, K)
        =sup_μ∈ K∫_{x≥ R}xμ(dx).


To summarize, the functions h,g are remarkably explicit and exactly mirror Eder's characterization of _1-relatively compact sets as stated in Proposition <ref>. In particular, g controls _1-relative compactness of K, while h controls the regularity of the kernels in K. 
The proof of the general result stated in Corollary <ref> essentially proceeds in the same fashion as described above, iterating over time periods t=1, …, T-1.




 §.§ Comparison of Wasserstein topologies 
 

To the best of our knowledge, general properties of the topology generated by _p^(σ) have not been studied. Given our explicit estimates in Theorems <ref> and <ref>, we can now address this question in great generality. To make our results concise, we consider a sequence (μ_n)_n∈ in _p((^d)^T) and an additional measure μ∈_p((^d)^T). We  refer to Section <ref> for exact statements.

Perhaps surprisingly, for fixed σ>0, the topology generated by ^(σ)_p is in fact the (vanilla) Wasserstein topology, i.e. 

    lim_n→∞^(σ)_p(μ,μ_n) =0   ⇔  lim_n→∞^(σ)_p(μ,μ_n) =0   ⇔  lim_n→∞_p(μ,μ_n) =0.

In this sense, all adapted topologies are equal as famously stated in <cit.>, but only if there is no uncertainty about the observed distributions μ, μ_n. As soon as these distributions are perturbed by some (arbitrarily small) independent noise ξ_σ, there is virtually no difference between the Wasserstein topology and the adapted Wasserstein topology. Phrased differently, in this case the finer structure of the adapted Wasserstein topology is absorbed by the noise ξ_σ.

The picture is different when one considers variable noise (ξ_σ_n)_n∈ instead. Now there is an interesting tradeoff between the size of the noise ξ_σ_n and the convergence speed of _1(μ,μ_n) on the one hand (cf. Theorem <ref>) and the regularity of kernels on the other hand (cf. Theorem <ref>). Figure <ref> illustrates this tradeoff in full generality. Here, the function h^t,p_μ_n is an iterated modulus of continuity, e.g., h^1,p_μ_n(σ_n)=ω^1,p_μ_n(σ_n), h^2,p_μ_n(σ_n)=ω^2,p_μ_n(σ_n+h^1,p_μ_n(σ_n)) and h^3,p_μ_n(σ_n)=ω^3,p_μ_n(σ_n+h^1,p_μ_n(σ_n)+h^2,p_μ_n(σ_n)). See Theorem <ref> for the precise definition of h^t,p_μ_n.






In the sense of Figure <ref>, convergence in ^(σ_n)_p is in general neither the same as convergence in _p nor convergence in _p and the smoothed adapted Wasserstein distance ^(σ_n)_p exhibits a non-trivial interpolation property between the different topologies. 
To motivate the above results, it is perhaps most instructive to return to the standard Example <ref>, where we can compute all relevant quantities explicitly. By convolving μ__n=1/2δ_(_n, 1)+1/2δ_(-_n, -1) with Gaussian noise ξ_σ_n=(0, σ^2_n_2), we see from Figure <ref> that kernels of μ__n are smoothed out. As _n→ 0, we can show that

    
  * _n/σ_n→ 0 if and only if _1(μ,μ__n)/σ_n→ 0 if and only if ^(σ_n)_p( μ, μ__n)→ 0, 
    
  * _n/σ_n→∞ if and only if ω^1,p_μ__n(σ_n)→ 0 if and only if _p(μ__n, μ__n∗ξ_σ_n)→ 0;

see Appendix <ref> for details. Roughly speaking, if σ_n decays slowly enough (so that _n/σ_n→ 0), the behavior of ^(σ_n)_p(μ, μ__n) mirrors the one of _p(μ, μ__n). Similarly, if σ_n decays fast enough (so that _n/σ_n→∞), we can use the triangle inequality to see that ^(σ_n)_p(μ, μ__n) and _p(μ, μ__n) converge toward the same value. In this sense, the tradeoff illustrated in Figure <ref> captures exactly the right scaling of _p(μ, μ__n) and σ_n, at least for the standard example.









 §.§ Related work
 

Adapted variants of the Wasserstein distance have been studied independently by many authors. In <cit.>, Aldous introduced the notion of extended weak convergence, which is essentially induced by the joint law between the original process and its corresponding prediction process. Further derivations of adapted topologies can be found in <cit.> and the references therein. Remarkably, <cit.> shows that all these seemingly different approaches define the same topology, which we refer to as the adapted Wasserstein topology throughout this paper. Based on the information topology proposed by <cit.>, Eder  characterizes the relatively compact sets in the adapted Wasserstein topology in <cit.>. The modulus of continuity proposed by Eder plays a central role in our work.

As discussed earlier, adapted transport maps induce bicausal couplings and as for their non-adapted counterparts, this inclusion is dense under mild regularity assumptions on the kernels <cit.>. The key idea is to embed bicausal couplings into enlarged spaces and approximate them by adapted maps. In Lemma <ref> we adapt these results to approximate a bicausal coupling between μ and μ∗ξ_σ by transport maps in order to bound the adapted Wasserstein distance between them. 

We also mention that duality of the adapted Wasserstein distance is investigated in <cit.>. Furthermore, <cit.> introduce a dynamic programming principle for the adapted Wasserstein distance, which we will use throughout this paper.

Notably, Eckstein–Pammer show in <cit.>, that the total variation distance and the adapted total variation distance (μ, ν):=inf_π∈(μ, ν)π({x≠ y}) are equivalent as metrics. To the best of our knowledge, this is the first result that gives upper and lower bounds of an adapted optimal transport cost by an optimal transport cost. Our work uses this estimate to bound _p in terms of _p.

Regarding applications, <cit.> show that many multistage optimization problems such as computing bid prices, are Hölder continuous with respect to _p. <cit.> extend this idea to a continuous time setting in mathematical finance and show that utility maximization, risk measurement and pricing/hedging problems are Lipschitz continuous with respect to _1.

Smoothing measures by convolving with Gaussian noise ξ is a standard tool in statistics, and is often used to obtain dimension-free empirical approximations. In optimal transport, the smooth Wasserstein distance and its concentration has been studied extensively.
We refer e.g. to <cit.> for both the slow rate [^(σ)_p(μ, μ_n)^p]≤ C/√(n) and the fast rate [^(σ)_p(μ, μ_n)]≤ C/√(n) under suitable moment conditions. Distributional limits of √(n)^(σ)_p(μ, μ_n) can be found in <cit.> for the case p=1 and in <cit.> for p>1.

It is well-known that the usual empirical measure μ_n is not adequate for empirical approximations in the adapted Wasserstein topology, i.e., lim_n→∞_p(μ,μ_n)≠ 0 in general. To alleviate this, <cit.> considers so-called adapted empirical measures for measures supported on compact sets, and prove the same dimension-dependent convergence rates as are known for their non-adapted counterparts. <cit.> extends this result to general measures. 

Quite recently, there have been  first efforts to study properties of the smooth adapted Wasserstein distance ^(σ)_p. Motivated by a martingale pair test, <cit.> obtain the  dimension-free convergence rate 1/√(n) for the so-called smoothed martingale projection distance. Under suitable moment assumptions, <cit.> establishes rates for [^(σ)_1(μ, μ_n)].

Organization of the paper
The rest of the paper is organized as follows. We end this introduction by setting up notation in Section <ref>.  In Section <ref> we present essential background materials for this paper, including the characterization of _p-relatively compact sets by <cit.> and basic properties of smooth Wasserstein distances. In Section <ref>, we provide the exact statements of the results that were briefly mentioned in Section <ref> and Section <ref>. Proofs of results in Section <ref> are contained in Section <ref>. Section <ref> is devoted to proofs of results in Section <ref>. In Appendix <ref>, the convergence results regarding Example <ref> are explained in detail. Appendix <ref> contains a denseness result used in the proof of Lemma <ref>.


Notation
As mentioned above, · is the Euclidean norm, and we denote the scalar (dot) product by ·. Throughout the paper,  T∈ is the number of time steps and d∈ is the dimension of the state space. The set of all Borel probability measures on (^d)^T  is denoted by ((^d)^T). For 1≤ p<∞, _p((^d)^T) is the set of all μ∈((^d)^T) that have finite p-moments. If μ∈((^d)^T), then we write M_p(μ):=∫_(^d)^Tx^p μ(dx) for the p-moment of μ. We denote the pushforward measure of μ under a Borel function g by g_#μ, i.e. g_#μ(A)=μ({x:  g(x)∈ A}) for all Borel sets A.

For x∈ (^d)^T, we use the shorthand notation x_t to denote the t-th coordinate of x and x_1:t:=(x_1, …, x_t) for t∈{1,2,…, T}. In particular, x=x_1:T. Analogously, for μ∈((^d)^T) and t∈{1,2,…, T}, the measure μ_t∈(^d) is the projection of μ onto the t-th coordinate, i.e., if P_t:(^d)^T→^d is defined as P_t(x_1, …, x_T)=x_t, then μ_t=(P_t)_#μ. Also, μ_1:t∈((^d)^t) is the pushforward measure of μ through the projection map P_1:t onto the first t coordinates, i.e., P_1:t(x_1, …, x_T)=(x_1, …, x_t). When integrating with respect to μ_t(dx_t) or μ_1:t(dx_1:t), we often drop subscripts and write

    ∫_^dg(x_t) μ_t(dx_t)
        =∫_^dg(x_t)μ(dx_t),  ∫_(^d)^tg(x_1:t) μ_1:t(dx_1:t)
        =∫_(^d)^tg(x_1:t)μ(dx_1:t)

if there is no confusion. Given x∈ (^d)^T we define μ_x_1:t∈(^d), t∈{1,2,…, T} via

    μ_x_1:t(dx_t+1)
        =(X_t+1∈ dx_t+1 X_1:t=x_1:t)

for X∼μ∈((^d)^T). 
In other words, (μ_x_1:t)_x_1:t∈ (^d)^t are the disintegrations of μ_1:t+1 along the first t coordinates.

Recall that (μ, ν) is the set of all couplings, (μ, ν) is the set of all bicausal couplings, the Wasserstein distance _p is defined in (<ref>) and _p in Definition <ref>. We adopt similar shorthand notation as above for π∈((^d)^T× (^d)^T): π_t,t∈(^d×^d) (respectively, π_1:t, 1:t∈((^d)^t× (^d)^t)) is the pushforward measure of π through the map (x_1, …, x_T, y_1, …, y_T)↦ (x_t, y_t) (respectively, (x_1, …, x_T, y_1, …, y_T)↦ (x_1, …, x_t, y_1, …, y_t)). We use the abbreviations π(dx_t, dy_t)=π_t,t(dx_t, dy_t) and π(dx_1:t, dy_1:t)=π_1:t, 1:t(dx_1:t, dy_1:t) as in (<ref>) if there is no confusion. Let us also remark that by definition we have

    π_x_1:t, y_1:t(dx_t+1, dy_t+1)
        =(X_t+1∈ dx_t+1, Y_t+1∈ dy_t+1 X_1:t=x_1:t, Y_1:t=y_1:t)

for (X, Y)∼π. 

Next we introduce the convolution operation ∗, i.e., given two probability measures μ, ν, we define μ∗ν(A):=∫μ(A-x)ν(dx) for all measurable A. In particular, if ν has density g, then μ∗ν has density μ∗ g(x):=∫ g(x-y)μ(dy). The (distributional) Fourier transform of μ∈((^d)^T), which we denote by μ: (^d)^T→, is given by μ(t)=∫_(^d)^T e^-2π i t· xμ(dx). For a function g, we similarly have g(t):=∫_(^d)^T e^-2π i t· xg(x)dx. Recall that μ∗ν=μν.

Unless otherwise stated, we reserve the notation ξ for the smoothing noise distribution. For σ>0, we write ξ_σ for the law of σ Z, where Z∼ξ. We use the shorthand notation μ^σ:=μ∗ξ_σ. See Section <ref> for details.

For any signed measure ρ on (^d)^T we define ρ_:=sup∑_j=1^∞ρ(E_j), where the supremum is taken over all measurable partitions (E_j)_j∈ of (^d)^T. Recall that if μ, ν∈((^d)^T),

    μ-ν_
        =2sup_G⊆ (^d)^T( μ(G)-ν(G))
        =2sup_π∈(μ, ν)π({x≠ y}).


Given μ∈^N and Σ∈^N× N, we write (μ, Σ) for the Gaussian distribution with mean vector μ and covariance matrix Σ. The N× N identity matrix is denoted as _N. In particular, (0, _N) is the standard Gaussian measure on ^N. 

We write C for a constant, that might change from line to line. We will detail the dependence of C on non-trivial parameters throughout the paper.







§ BACKGROUND

In this section, we review some auxiliary results on characterizations of relative compactness in Wasserstein distance and smooth Wasserstein distances. Proofs of all results in this section can be found in Section <ref>.



 §.§ Compactness in the adapted Wasserstein topology

Recall that _p was introduced in (<ref>) and _p was introduced in Definition <ref>. Given x≤ C (∑_t=1^T x_t^p)^1/p, where C=1 if 1≤ p≤ 2 and C=T^1/2-1/p if p≥ 2, we have _p≤ C_p. Hence, the _p-relatively compact subsets of _p((^d)^T) are also _p-relatively compact. In fact, Eder showed in <cit.> that the _p-relatively compact sets are exactly those _p-relatively compact sets whose modulus of continuity converges uniformly.

    [Modulus of continuity, <cit.>]
        Let μ∈_p((^d)^T). The p-modulus of continuity of μ, denoted by ω_μ^t,p:(0,∞)→ [0, ∞), t∈{1,2,…, T-1}, is given by
        
    ω_μ^t,p(δ)
                =sup{([_p(μ_X_1:t, μ_Y_1:t)^p])^1/p |  X, Y∼μ, ([X_1:t-Y_1:t^p])^1/p<δ}.

    

<ref>
    
        Let K⊆_p((^d)^T). Then K is _p-relatively compact if and only if K is _p-relatively compact and
        
    lim_δ→ 0sup_μ∈ Kω^t,p_μ(δ)=0
                   for all  t∈{1,2,…, T-1}.

    
   thm-1
The characterization of compactness in Proposition <ref> is slightly different from the original result by Eder <cit.>. However, it is not hard to see that they are equivalent, as we prove in Section <ref>.
    
        Trivially K={μ} is _p-relatively compact. Hence, ω^t,p_μ(δ)→ 0 as δ→ 0.
    
    
        Recall μ_=1/2δ_(-, -1)+1/2δ_(, 1) from Example <ref>. Note that K:={μ_: 0<<1} is _p-relatively compact. However, (μ_)_x=δ_1 if x= and (μ_)_x=δ_-1 if x=-. Thus we easily compute that ω_μ_^1,p(δ)=(δ/)∧ 2 and sup_0<<1ω_μ_^1,p(δ)=2. Proposition <ref> implies that K is not _p-relatively compact.
     
Next we present some basic properties of the modulus of continuity ω.
    
        Let μ∈_p((^d)^T) and t∈{1,2,…, T-1}.
        
            
  *  The function ω^t,p_μ is monotone increasing, i.e. ω^t,p_μ(δ_1)≤ω^t,p_μ(δ_2) for δ_1<δ_2.
            
  *  For k, δ>0, ω^t,p_μ(kδ)≤ (k∨ 1)ω^t,p_μ(δ).
        
    
    If the kernels (^d)^t∋ x_1:t↦μ_x_1:t∈_p(^d) are equicontinuous,  where _p(^d) is endowed with a metric _p, then the modulus of continuity of μ converges uniformly. An important class of meausres with equicontinuous kernels is given by the ones with  Hölder continuous kernels.
    
        Let L>0 and 0<α≤ 1. The set _p, α, L((^d)^T) contains all μ∈_p((^d)^T) that satisfy
        
    _p(μ_x_1:t, μ_y_1:t)≤ L x_1:t-y_1:t^α

        for all x_1:t, y_1:t∈ (^d)^t and t∈{1, 2, …, T-1}.
    

    
        Let L>0 and 0<α≤ 1. If μ∈_p, α, L((^d)^T), then ω^t,p_μ(δ)≤ L δ^α for δ>0 and t∈{1,2,…, T-1}.
    
   
What is more, measures with Lipschitz kernels (i.e. α=1) arise in many practical examples. We refer to <cit.> for some simple sufficient conditions.
    




 §.§ Smooth adapted Wasserstein distances

As explained in Section <ref>, we define the smooth (adapted) Wasserstein distance as follows.
    [Smooth distances]
        Let ξ∈_p((^d)^T). For σ>0 and Z∼ξ, denote by ξ_σ the law of σ Z. We define the smooth p-Wasserstein distance with the smoothing parameter σ via
        
    ^(σ)_p(μ, ν)
                :=_p(μ∗ξ_σ, ν∗ξ_σ)  for μ, ν∈_p((^d)^T).

        Similarly, the smooth p-adapted Wasserstein distance with the smoothing parameter σ is given by
        
    ^(σ)_p(μ, ν)
                :=_p(μ∗ξ_σ, ν∗ξ_σ)  for μ, ν∈_p((^d)^T).

    
    
        For μ∈((^d)^T), we often use the shorthand notation μ^σ=μ∗ξ_σ. In particular, ^(σ)_p(μ, ν)=_p(μ^σ, ν^σ) and ^(σ)_p(μ, ν)=_p(μ^σ, ν^σ).
    
    In the following proposition, we present some properties of the smooth Wasserstein distance ^(σ)_p. While these results are simple extensions of the classical theory, we provide proofs in Section <ref> to keep our work self-contained.
    
    
        Suppose that the set of real zeros of ξ has Lebesgue measure zero. Then, for σ>0, ^(σ)_p is a metric on _p((^d)^T) that induces the same topology as _p. Furthermore, if σ_1, σ_2>0, then
        
    |^(σ_1)_p(μ, ν)-^(σ_2)_p(μ, ν)|
                ≤ 2σ_1-σ_2(M_p(ξ))^1/p

        for μ, ν∈_p((^d)^T). Here, recall that M_p(ξ)=∫_^dTx^p ξ(dx).
    
    The assumption that ξ does not admit any real zeros a.s. is essential to guarantee that ^(σ)_p defines a metric. This is satisfied, for example, when ξ is compactly supported (see <cit.>) or when ξ is Gaussian. The same result is true for ^(σ)_p.
    
    
        Suppose that the set of real zeros of ξ has Lebesgue measure zero. Then _p^(σ) is a metric on _p((^d)^T) for σ>0.
    




§ MAIN RESULTS

Now, we have all the necessary ingredients to present the main results. Proofs for this section can be found in Section <ref>. We begin with some standing assumptions on the  noise measure ξ. 
    
 We have the following:
        
            
  *  The noise distribution ξ∈_p((^d)^T)  has density f:(^d)^T→ which is continuously differentiable and f∈ L^1((^d)^T; (^d)^T). In particular, for σ>0, we write  f_σ(·):=σ^-dTf(·/σ) for the density of ξ_σ.
            
  *  The set of real zeros of f (or equivalently ξ) has Lebesgue measure zero.
        
    
    Assumption <ref> is a mild growth condition on f. Note from Theorem <ref> that <ref> guarantees that ^(σ)_p (as well as ^(σ)_p) is indeed a metric on the space of probability measures.

    

 §.§ The adapted Wasserstein distance and the total variation norm

     Recall that the p-Wasserstein distance _p(μ, ν) can be controlled by the p-total variation norm, ·^p(μ-ν)_. For example, <cit.> shows that
     
    _p(μ, ν)
             ≤ 2^1-1/p(∫x^p μ-ν(dx))^1/p.

     Similarly, Proposition <ref> shows that _p is bounded by the total variation norm ·_ plus remainder terms that can be made arbitrarily small. This result is based on <cit.> stating that the metric (μ, ν)=inf_π∈(μ, ν)π({x≠ y}) is equivalent to the total variation distance 1/2μ-ν_=inf_π∈(μ, ν)π({x≠ y}).
    
    Let μ, ν∈_p((^d)^T). For any R>0 we have
        
    _p(μ, ν)^p
                ≤ 6^p-1T2^T (R^p μ-ν_
                +∫_{x≥ R}x^p (μ+ν)(dx)).

    
    Our first main result, Theorem <ref>, asserts that the smooth p-adapted Wasserstein distance ^(σ)_p can be bounded by the 1-Wasserstein distance _1. 
    
    Let μ, ν∈_p((^d)^T). For any R>0 and σ>0 we have
        
    ^(σ)_p(μ, ν)^p
                ≤ 6^p-1T2^T (R^p f_L^1_1(μ, ν)/σ
                +∫_{x≥ R}x^p (μ^σ+ν^σ)(dx)).

    
    Under additional assumptions on the measures μ and ν, we can actually drop the additive terms characterizing the tail behavior.

    
        For any σ>0 we have the following:
        
            
  *  If μ, ν, ξ∈_q((^d)^T) for some q>p, then
            
    ^(σ)_p(μ, ν)^p
                    ≤ 6^p-1T2^T+1(M_q(μ^σ+ν^σ))^p/q( f_L^1_1(μ, ν)/σ)^1-p/q.

            
  *  If μ, ν and ξ have bounded supports, then
            
    ^(σ)_p(μ, ν)^p
                     ≤ 6^p-1T2^T+2(((μ)∪(ν))+σ((ξ)))^p/σ f_L^1_1(μ, ν).

            
  *  Suppose ξ∼(0, _dT). If μ, ν∈_q((^d)^T) for some q>p, then for 0<σ_0<σ,
            
    ^(σ)_p(μ, ν)^p
                    ≤ 6^p-1T2^T+1 f^1-p/q_L^1(M_q(μ^σ+ν^σ))^p/q(^(σ_0)_1(μ, ν)/√(σ^2-σ^2_0))^1-p/q.

        
    

    Combining Theorem <ref> with the slow rate of ^(σ)_1 yields the following upper bound on the convergence rate between the true and empirical measure in ^(σ)_p. Similar bounds were announced in <cit.>.

     Let μ∈_q((^d)^T) for some q>p∨ (dT+2). Suppose ξ∼(0, _dT). Denote by μ_n the empirical measure of μ with sample size n∈, i.e. μ_n=1/n∑_j=1^n δ_X^(j) where X^(1), X^(2), …, X^(n) are i.i.d samples of μ. Then
        
    [^(σ)_p(μ, μ_n)^p]
                ≤ C(1+σ^(q+p)/2/σ^(1+dT/2)(1-p/q)) n^-(q-p)/(2q)

        for some constant C>0 that depends only on d, T, p, q, M_q(μ).
    

    
        If μ has a bounded support, then we have
        
    [^(σ)_p(μ, μ_n)^p]≤ C n^-1/2+

        for all >0. This recovers the slow rate [^(σ)_p(μ, μ_n)^p]≤ Cn^-1/2 up to an -loss.
    
   

    

 §.§ Smoothing and the modulus of continuity

    Our second main result pertains to the bandwidth effect of μ^σ. As shown in <cit.>, the smoothing operation is stable in the sense that _p(μ, μ^σ)→ 0 as σ→ 0. Here, we establish that convergence is controlled by the modulus of continuity defined in Definition <ref>.
        
     Let μ∈_p((^d)^T). Then for any σ>0,
        
    _p(μ, μ^σ)
                ≤ T(1∨(M_p(ξ))^1/p)∑_t=0^T-1h^t,p_μ(σ)

        where h^t,p_μ is defined as follows
        
    h^0,p_μ(σ)=σ,    
                h^t,p_μ(σ)=ω_μ^t,p(∑_s=0^t-1 h^s,p_μ(σ))  for  t∈{1,2, …, T-1}.

    

    
         The following hold.
        
            
  *  For L>0 and 0<α≤ 1, there exists a constant C>0 that depends only on T, (M_p(ξ))^1/p, L, α such that
            
    sup_μ∈_p, α, L((^d)^T)_p(μ, μ^σ)
                    ≤ C ∑_t=0^T-1σ^α^t.

            
  *  Let K⊆_p((^d)^T). If sup_μ∈ Kω^t,p_μ(δ)→ 0 as δ→ 0 for all t∈{1,2,…, T-1}, then
            
    lim_σ↘ 0sup_μ∈ K_p(μ, μ^σ)=0.

            In particular,
            
    lim_σ↘ 0sup_μ, ν∈ K^(σ)_p(μ, ν)-_p(μ, ν)
                    =0.

        
    

    By choosing a suitably decaying sequence (σ_n)_n∈, ((μ_n)^σ_n)_n∈ approaches μ in _p. The convergence rate depends on the dimension dT.
    
    
        Let μ∈_q((^d)^T)∩_p, α, L((^d)^T) for some q>p∨ (dT+2), L>0 and 0<α≤ 1. Suppose ξ∼(0, _dT). Using the same notation as in Corollary <ref>, there exists a constant C>0 that depends only on d, T, p, q, M_q(μ), L, α such that
        
    [_p(μ, (μ_n)^σ_n)^p]
                ≤ C n^-β

        where
        
    σ_n = n^-1-p/q/(dT+2)(1-p/q)+2pα^T-1 and β = pα^T-1(1-p/q)/(dT+2)(1-p/q)+2pα^T-1.

    


    

 §.§ TEXT

    Now, we turn our attention to bounding _p in terms of _1. By introducing an auxiliary noise ξ satisfying Assumption <ref>, a straightforward application of the triangle inequality to Theorem <ref> and Theorem <ref> shows the following proposition.
    
        Let μ, ν∈_p((^d)^T). Then for any R>0, σ>0 and ξ satisfying Assumption <ref> we have
        
    _p(μ, ν)^p
                   ≤ 18^p-1T2^T R^p f_L^1_1(μ, ν)/σ
                +18^p-1T2^T∫_{x≥ R}x^p (μ^σ+ν^σ)(dx)
       +3^p-1T^p(1∨ M_p(ξ))(∑_t=0^T-1h^t,p_μ(σ))^p
                +3^p-1T^p(1∨ M_p(ξ))(∑_t=0^T-1h^t,p_ν(σ))^p.

    
    We remark that R>0, σ>0 and ξ (or equivalently f) can be chosen arbitrarily in the above, as long as Assumption <ref> is satisfied. We present several estimates as outcomes of specific choices of R, σ or ξ.

    
        Let K⊆_p((^d)^T) be _p-relatively compact. Then there exists a constant C>0 that depends only on d, T, p such that for all 0<σ<R and μ, ν∈ K,
        
    _p(μ, ν)^p
                ≤ C(R^p _1(μ, ν)/σ
                +sup_γ∈ K(∑_t=0^T-1h^t,p_γ(σ))^p
                +sup_γ∈ K∫_{x≥ R}x^p γ(dx)).

    

    
        Recall from Proposition <ref> that the relative compactness of K yields
        
    lim_σ↘ 0sup_γ∈ K∑_t=0^T-1h^t,p_γ(σ)=0,  lim_R→∞sup_γ∈ K∫_{x≥ R}x^p γ(dx)=0.

    

    
    Let μ, ν∈(F) for some bounded set F⊆ (^d)^T.
    
        
  *  There exists a constant C>0 that depends only on d, T, p, (F) such that for all σ>0,
        
    _p(μ, ν)^p
                ≤ C((1+σ)^p/σ_1(μ, ν)+(∑_t=0^T-1h^t,p_μ(σ)∨ h^t,p_ν(σ))^p).

        
  *  Further suppose that μ, ν∈_p, α, L((^d)^T) for some L>0 and 0<α≤ 1. Then there exists a constant C>0 that depends only on d, T, p, (F), L, α such that
        
    _p(μ, ν)
                ≤ C_1(μ, ν)^α^T-1/(pα^T-1+1).

    
    


    

 §.§ Comparison of topologies

    In this section, we complement the discussion in Section <ref> and Figure <ref> by giving concise mathematical statements.


     Let  μ∈_p((^d)^T) and (μ_n)_n∈⊆_p((^d)^T). Consider (σ_n)_n∈⊆ (0,∞) such that sup_n∈σ_n<∞.
    
        
  *  If ^(σ_n)_p(μ_n, μ)→ 0, then _p(μ_n, μ)→ 0. 
        
  *  If _p(μ_n, μ)→ 0 and _1(μ, μ_n)/σ_n→ 0, then ^(σ_n)_p(μ_n, μ)→ 0.
    
    

     Let  μ∈_p((^d)^T) and (μ_n)_n∈⊆_p((^d)^T). Consider (σ_n)_n∈⊆ (0,∞) such that sup_n∈σ_n<∞.
    
        
  *  If _p(μ_n, μ)→ 0, then ^(σ_n)_p(μ_n, μ)→ 0. 
        
  *  If ^(σ_n)_p(μ_n, μ)→ 0 and h^t,p_μ_n(σ_n)→ 0 for all t∈{1,2,…, T-1}, then _p(μ_n, μ)→ 0.
    
    

    
        Let σ>0. Then ^(σ)_p is a metric on _p((^d)^T) that induces the same topology as _p.
    


In conclusion, adding independent noise ξ fundamentally changes the temporal structure of stochastic processes and makes the smooth adapted Wasserstein topology different from the adapted Wasserstein topology in general. This contrasts with the Wasserstein topology, where the smooth Wasserstein distance generates the Wasserstein topology. In the case where the smoothing parameter is fixed for some σ>0, the smooth adapted Wasserstein distance completely ignores temporal evolution captured by the natural filtration.





§ PROOFS FOR SECTION <REF>


In this section we give proofs for the results of Section <ref>. Proofs for Section <ref> are contained in Section <ref> and proofs for Section <ref> can be found in Section <ref>.



 §.§ Proofs for Section <ref>


In this section, we first state and prove Lemma <ref>, which is essential for proving Proposition <ref>. We then present proofs of Proposition <ref>, Proposition <ref> and Proposition <ref>. We start by establishing some notation that will be used exclusively in the proofs of Lemma <ref> and Proposition <ref>.

Let μ∈_p((^d)^T). For t∈{1,2,…, T-1} and x_1:t∈ (^d)^t, we define μ̅_x_1:t∈_p((^d)^T-t) via

    μ̅_x_1:t(dx_t+1:T)
        =μ_x_1:T-1(dx_T)⋯μ_x_1:t+1(dx_t+2)μ_x_1:t(dx_t+1).

Equivalently, when X∼μ and A⊆ (^d)^T-t is measurable,

    μ̅_x_1:t(A)
        =(X_t+1:T∈ A  X_1:t=x_1:t).

Using this, we define ω̅_μ^t,p:(0,∞)→ [0,∞) via

    ω̅_μ^t,p(δ)
        =sup{([_p(μ̅_X_1:t, μ̅_Y_1:t)^p])^1/p |  X, Y∼μ, ([X_1:t-Y_1:t^p])^1/p<δ}.


        

    Let μ∈_p((^d)^T). For δ>0 and t∈{1,2,…, T-1}, we have
    
    ω̅_μ^t,p(δ)
            ≤∑_s=t^T-1 g_t,μ^s,p(δ)

    where
    
    g_t,μ^s,p(δ)
            :=ω_μ^s,p(δ+∑_ℓ=t^s-1 g_t,μ^ℓ, p(δ))  for  s∈{t,t+1 …, T-1}.

    Here, the sum ∑_ℓ=t^s-1 is understood as 0 when s=t.


    Let us fix t∈{1,2,…, T-1} and π∈(μ_1:t, μ_1:t) such that
    
    (∫x_1:t-y_1:t^p π(dx_1:t, dy_1:t))^1/p<δ.

    For s∈{t, t+1, …, T-1} and x_1:s, y_1:s∈ (^d)^s, define π^⋆_x_1:s, y_1:s∈(μ_x_1:s, μ_y_1:s) as a _p-optimal coupling between μ_x_1:s and μ_y_1:s. Note that γ∈(μ, μ) which is defined via
    
    γ(dx_1:T, dy_1:T)
       :=π^⋆_x_1:T-1, y_1:T-1(dx_T, dy_T)⋯π^⋆_x_1:t+1, y_1:t+1(dx_t+2, dy_t+2)π^⋆_x_1:t, y_1:t(dx_t+1, dy_t+1)π(dx_1:t, dy_1:t).

    Let (X, Y)∼γ. From the construction of γ, we bound
    
    ([X_s+1-Y_s+1^p])^1/p   =(∫x_s+1-y_s+1^p γ(dx_1:s+1, dy_1:s+1))^1/p
       =(∫_p(μ_x_1:s, μ_y_1:s)^p γ(dx_1:s, dy_1:s))^1/p
       ≤ω_μ^s, p(([X_1:s-Y_1:s^p])^1/p)
       ≤ω_μ^s,p(δ+∑_ℓ=t^s-1([X_ℓ+1-Y_ℓ+1^p])^1/p).

    Here the first inequality comes from the definition of ω_μ^s,p and the last inequality is due to the inequality (<ref>),
    
    ([X_1:t-Y_1:t^p])^1/p
            =(∫x_1:t-y_1:t^p π(dx_1:t, dy_1:t))^1/p
            <δ.

    Using the recursive formula (<ref>), we obtain that ([X_s+1-Y_s+1^p])^1/p≤ g_t,μ^s,p(δ). Observe that
    
    (∫_p(μ̅_x_1:t, μ̅_y_1:t)^p π(dx_1:t, dy_1:t))^1/p   ≤([X_t+1:T-Y_t+1:T^p])^1/p
       ≤∑_s=t^T-1([X_s+1-Y_s+1^p])^1/p≤∑_s=t^T-1g_t,μ^s,p(δ).

    Since the above estimate (<ref>) holds for all π∈(μ_1:t, μ_1:t) satisfying (<ref>), taking the supremum over all such π in the left hand side of (<ref>) gives the desired estimate.


    
    
        It follows from <cit.> that K⊆_p((^d)^T) is _p-relatively compact if and only if K is _p-relatively compact and
        
    lim_δ→ 0sup_μ∈ Kω̅_μ^t,p(δ)=0  for all  t∈{1,2,…, T-1}

        It is straightforward from the inequality _p(μ_x_1:t, μ_y_1:t) ≤_p(μ̅_x_1:t, μ̅_y_1:t) for all x_1:t, y_1:t∈ (^d)^t, that condition (<ref>) implies condition (<ref>). The fact that condition (<ref>) implies condition (<ref>) follows from Lemma <ref>.
    


    
    <ref> The proof is obvious from the definition of ω^t,p_μ.
    
    <ref>
        If k≤ 1, the proof is immediate from the fact that a map ω_μ^t,p(·) is nondecreasing. Now let us assume k>1. The proof for the case k>1 is a simple re-scaling of <cit.>. For the sake of simplicity, let us write =(^d)^t, =(^d) and define a map Ł_t:((^d)^T)→(×) via Ł_t(ν)=(Y_1:t, ν_Y_1:t) for Y∼ν. As a byproduct of <cit.>, we can relax the marginal constraints as follows:
        
    ω_μ^t,p(δ)
                =sup_γ∈(Ł_t(μ), δ)ρ^(γ)

        where
        
    ρ^(γ)=(∫x_1:t-y_1:t^p γ(dx_1:t, dκ, dy_1:t, dρ))^1/p,
       ρ^(γ)=(∫_p(κ, ρ)^p γ(dx_1:t, dκ, dy_1:t, dρ))^1/p,
       (Ł_t(μ), δ)
                ={γ∈^≤(×××) : ρ^(γ)<δ, P^(1)_#γ≤Ł_t(μ), P^(2)_#γ≤Ł_t(μ)}.

        Here, ^≤(×××) is the collection of positive measures whose total mass is no greater than 1 and the maps P^(j):×××→× are defined via P^(j)(x_1, y_1, x_2, y_2):=(x_j, y_j) for j∈{1,2}. Hence if γ∈(Ł_t(μ), kδ), then k^-pγ∈(Ł_t(μ), δ) and ρ^(γ)=kρ^(k^-pγ). This implies that ω_μ^t,p(kδ)≤ kω_μ^t,p(δ).
    

    
        Let μ∈_p, α, L((^d)^T). Suppose X, Y∼μ and [X_1:t-Y_1:t^p]<δ^p. Applying Jensen's inequality we obtain
        
    ([_p(μ_X_1:t, μ_Y_1:t)^p])^1/p≤ L([X_1:t-Y_1:t^pα])^1/p≤ L ([X_1:t-Y_1:t^p])^α/p≤ L δ^α.

        Hence, ω^t,p_μ(δ)≤ L δ^α.
    



 §.§ Proofs for Section <ref>

    
         The proof is a straightforward generalization of <cit.>: the case where the noise ξ is Gaussian is treated in <cit.> and <cit.> shows the same result when ξ is compactly supported.
         
         The fact that ^(σ)_p and ^(σ)_p satisfy symmetry and the triangle inequality is trivial; for example, see <cit.>. If ^(σ)_p(μ, ν)=0 or ^(σ)_p(μ, ν)=0, then μ∗ξ_σ=ν∗ξ_σ. By taking Fourier transforms, we deduce that
         
    μ(t)ξ(σ t)
                 =μ∗ξ_σ(t)
                 =ν∗ξ_σ(t)=ν(t)ξ(σ t)  for all  t∈ (^d)^T.

         Since the set of real zeros of ξ has Lebesgue measure zero, μ=ν a.s. In particular, μ=ν as tempered distributions. The Fourier inversion formula shows that μ=ν as tempered distributions and thus μ=ν as measures. This shows that ^(σ)_p and ^(σ)_p are metrics on _p((^d)^T). For a proof that ^(σ)_p metrizes the _p-topology we refer to <cit.>. Lastly, for σ_1, σ_2>0, find (X, Z)∼μ⊗ξ and (Y, W)∼ν⊗ξ such that (X+σ_1 Z, Y+σ_1 W) is an optimal coupling of ^(σ_1)_p(μ, ν). Then by the triangle inequality,
         
    ^(σ_2)_p(μ, ν)
                    ≤([(X+σ_2 Z)-(Y+σ_2 W)^p])^1/p
       ≤([(X+σ_1 Z)-(Y+σ_1 W)^p])^1/p
                 +σ_1-σ_2([Z-W^p])^1/p
       ≤^(σ_1)_p(μ, ν)+2σ_1-σ_2(M_p(ξ))^1/p.

         This proves desired estimates.
    








§ PROOFS FOR SECTION <REF>

Proofs for Sections <ref>, <ref>, <ref> and <ref> are contained in Sections <ref>, <ref>, <ref> and <ref>.



 §.§ Proofs for Section <ref>

This section is organized as follows. In Lemma <ref>, we show that compactly supported measures are _p-dense via an explicit estimate. Combined with Lemma <ref>, this proves Proposition <ref>. The proof of Theorem <ref> is based on Lemma <ref>. Following that, we prove Corollary <ref> and Corollary <ref>.


    Let R>0. There exists a Lipschitz function Φ^(R)=(Φ^(R)_1, …, Φ^(R)_T):(^d)^T→ (^d)^T such that
    
    _p(μ, Φ^(R)_#μ)
            ≤ 2T^2/p(∫_{x≥ R}x^p μ(dx))^1/p

    for any μ∈_p((^d)^T). Moreover, Φ^(R)_t≤ R for every t∈{1,2,…, T}.


    Throughout the proof, we assume that R>0 and μ∈_p((^d)^T) are fixed. We define Φ:(^d)^T→ (^d)^T via Φ(x_1:T):=(φ(x_1), …, φ(x_T)), where φ:^d→^d is given by
    
    φ(x)
            =
                x     if x< R
    Rx/x    if x≥ R.

    We want to construct a bicausal coupling between μ and Φ_#μ. Set π_1,1:=(𝕀, φ)_#μ_1∈(ν_1, (Φ_#ν)_1). For t∈{1,2,…, T-1} and x_1:t, y_1:t∈ (^d)^t, let us define π_x_1:t, y_1:t∈(^d×^d) via
    
    π_x_1:t, y_1:t
            :=
                (𝕀, φ)_#μ_x_1:t    if  x_1:t=y_1:t and x_ℓ<R  for all ℓ∈{1,2,…, t},
    μ_x_1:t⊗ (Φ_#μ)_y_1:t    otherwise.

    Since (Φ_#μ)_y_1:t=φ_#μ_x_1:t on {x_1:t=y_1:t and x_ℓ<R  for all ℓ∈{1,2,…, t}}, we have that π_x_1:t, y_1:t∈(μ_x_1:t, (Φ_#μ)_y_1:t). Define the measure π∈((^d)^T× (^d)^T) via 
    
    π(dx, dy)
            :=π_x_1:T-1, y_1:T-1(dx_T, dy_T)⋯π_x_1, y_1(dx_2, dy_2)π_1,1(dx_1, dy_1)

    and conclude that π∈(μ, Φ_#μ). Thus
    
    _p(μ, Φ_#μ)^p
            ≤∑_t=1^T∫x_t-y_t^p π(dx_1:t, dy_1:t).

    For t∈{1, 2, …, T}, consider the partition (E^t_ℓ)_ℓ=0^t of (^d)^t× (^d)^t where
    
    E^t_0:={(x_1:t, y_1:t): x_s<R  for all  s∈{1,2, …, t}},
       E^t_ℓ
            :={(x_1:t, y_1:t): x_ℓ≥ R  and x_s< R  for all  s∈{1,2, …, ℓ}} when ℓ∈{1,2,…, t}.

    Note that
    
    ∫_E^t_0x_t-y_t^p π(dx_1:t, dy_1:t)
            =∫_{x_t<R }x_t-φ(x_t)^p μ(x_1:t)=0.

    For ℓ∈{1,2, …, t}, we use Jensen's inequality to compute
    
    ∫_E^t_ℓx_t-y_t^p π(dx_1:t, dy_1:t)
               ≤ 2^p-1∫_E^t_ℓx_t^p μ(dx_1:t)
            +2^p-1∫_E^t_ℓy_t^p Φ_#μ(dy_1:t)
       ≤ 2^p∫_{x_ℓ≥ R}x_t^p μ(dx_1:t).

    Combining these results,
    
    _p(μ, Φ_#μ)^p
               ≤∑_t=1^T ∑_ℓ=1^t∫_E^t_ℓx_t-y_t^p π(dx_1:t, dy_1:t)
       ≤∑_t=1^T ∑_ℓ=1^t
            2^p∫_{x_ℓ≥ R}x_t^p μ(dx_1:t)
            ≤ 
            2^pT^2∫_{x≥ R}x^p μ(dx).



    
    For μ, ν∈((^d)^T) we have
    
    1/2μ-ν_≤(μ, ν)
            :=inf_π∈(μ, ν)π({x≠ y})
            ≤2^T-1/2μ-ν_.



    See <cit.>.


    
        Fix R>0 and define Φ^(R) as in Lemma <ref>. Observe that
        
    _p(Φ^(R)_#μ, Φ^(R)_#ν)^p
                   ≤(2R)^pinf_π∈(Φ^(R)_#μ, Φ^(R)_#ν)∫∑_t=1^T {x_t≠ y_t}π(dx_1:T, dy_1:T)
       ≤ (2R)^p T inf_π∈(Φ^(R)_#μ, Φ^(R)_#ν)∫{x_1:T≠ y_1:T}π(dx_1:T, dy_1:T)
       = (2R)^p T (Φ^(R)_#μ, Φ^(R)_#ν)
       ≤ (2R)^p T((2^T-1)/2) Φ^(R)_#μ-Φ^(R)_#ν_
       ≤ (2R)^p T((2^T-1)/2) μ-ν_.

        The first inequality stems from the inclusion {x_t≠ y_t}⊆{x_1:T≠ y_1:T}. Lemma <ref> gives the second inequality. The last inequality is straightforward from the definition of ·_. Indeed,
        
    1/2Φ^(R)_#μ-Φ^(R)_#ν_   =sup_A⊆ (^d)^Tμ((Φ^(R))^-1(A))-ν((Φ^(R))^-1(A))
       ≤sup_B⊆ (^d)^Tμ(B)-ν(B)
                =1/2μ-ν_.

        Using the triangle inequality together with the estimate (<ref>) and Lemma <ref>,
        
    _p(μ, ν)^p
                   ≤ 3^p-1(_p(μ, Φ^(R)_#μ)^p+_p(ν, Φ^(R)_#ν)^p+_p(Φ^(R)_#μ, Φ^(R)_#ν)^p)
       ≤ 3^p-12^p T^2 ∫_{x≥ R}x^p (μ+ν)(dx)
                +3^p-1(2R)^p T((2^T-1)/2) μ-ν_
       ≤ (3^p-12^p T^2)∨ (3^p-12^p T((2^T-1)/2)) (∫_{x≥ R}x^p (μ+ν)(dx)
                +R^pμ-ν_)
       ≤ 6^p-1T2^T(∫_{x≥ R}x^p (μ+ν)(dx)+R^pμ-ν_).

        This proves the desired estimate.
    
    

    
        For μ, ν∈_p((^d)^T) we have
        
    μ^σ-ν^σ_≤ f_L^1_1(μ, ν)/σ.

    
    
        We first claim that
        
    μ^σ-ν^σ_   =sup{∫φ dμ^σ-∫φ dν^σ : φ≤ 1}
       =sup{∫φ dμ^σ-∫φ dν^σ : φ≤ 1, ((φ))<∞}.

        The first equality is classical, see for example <cit.>. To prove the second equality, let us fix >0 and choose R>0 such that (μ^σ+ν^σ)({x∈ (^d)^T: x>R})≤. Take any function φ:(^d)^T→ such that φ≤ 1, set φ_R(x):=φ(x){x≤ R} and observe that
        
    ∫φ dμ^σ-∫φ dν^σ   =(∫φ_R dμ^σ-∫φ_R dν^σ)
                +(∫ [φ-φ_R] dμ^σ-∫ [φ-φ_R] dν^σ)
       ≤(∫φ_R dμ^σ-∫φ_R dν^σ)
                +.

        Since φ is arbitrary, it shows the claim.

        Now, let φ:(^d)^T→ such that φ≤ 1 and ((φ))<∞. Note that
        
    ∫φ dμ^σ-∫φ dν^σ
                =∫(∫φ(x)f_σ(x-y)dx)μ(dy)-∫(∫φ(x)f_σ(x-y)dx)ν(dy).

        Since φ has bounded support and f is continuously differentiable, the dominated convergence theorem implies that
        
    _y(∫φ(x)f_σ(x-y)dx)
                =-∫φ(x) ( f_σ)(x-y)dx.

        In particular
        
    _y(∫φ(x)f_σ(x-y)dx)≤∫ f_σ(x)dx
                =1/σ∫ f(x)dx.

        Thus, the function y↦∫φ(x)f_σ(x-y)dx is Lipschitz continuous with the Lispchitz constant f_L^1/σ. The desired estimate follows from the Kantorovich–Rubinstein duality for _1(μ, ν) and (<ref>).
    

    
    Combining Proposition <ref> and Lemma <ref>,
        
    _p(μ^σ, ν^σ)^p
                   ≤ 6^p-1T2^T (R^p μ^σ-ν^σ_
                +∫_{x≥ R}x^p (μ^σ+ν^σ)(dx))
       ≤ 6^p-1T2^T (R^p  f_L^1_1(μ, ν)/σ
                +∫_{x≥ R}x^p (μ^σ+ν^σ)(dx)).

    

    
        <ref> For the sake of simplicity, let us write A_R:={x∈ (^d)^T : x≥ R}. Using the Markov inequality, we bound
        
    ∫_{x≥ R}x^p (μ^σ+ν^σ)(dx)
                   =∫x^p A_R(x)(μ^σ+ν^σ)(dx)
       ≤∫x^p x^q-p/R^q-p(μ^σ+ν^σ)(dx)
                =M_q(μ^σ+ν^σ)/R^q-p.

        Plugging this into Theorem <ref>,
        
    ^(σ)_p(μ, ν)^p
                ≤ 6^p-1T2^T (R^p f_L^1_1(μ, ν)/σ
                +M_q(μ^σ+ν^σ)/R^q-p).

        The desired estimate (<ref>) follows from optimizing the right-hand side by choosing
        
    R=(σ M_q(μ^σ+ν^σ)/ f_L^1_1(μ, ν))^1/q.


        <ref> Without loss of generality, we may assume that 0∈ ((μ)∪(ν))∩(ξ). The proof for the general case follows from translations. Applying the bound
        
    (M_q(μ^σ+ν^σ))^p/q   ≤ (M_q(μ^σ))^p/q
                +(M_q(ν^σ))^p/q
       ≤ ((M_q(μ))^1/q+σ (M_q(ξ))^1/q)^p
                +((M_q(ν))^1/q+σ (M_q(ξ))^1/q)^p
       ≤ 2(((μ)∪(ν))+σ((ξ)))^p

        and sending the limit q→∞ in (<ref>), we obtain
        
    ^(σ)_p(μ, ν)^p
                   ≤ 6^p-1T2^T+1lim inf_q→∞(M_q(μ^σ+ν^σ))^p/q( f_L^1_1(μ, ν)/σ)^1-p/q
       ≤ 6^p-1T2^T+2(((μ)∪(ν))+σ((ξ)))^p( f_L^1_1(μ, ν)/σ).


        <ref> If ξ is a Gaussian measure we have ξ_σ=(ξ_σ_0)∗ (ξ_√(σ^2-σ^2_0)). Since q>p, we apply Corollary <ref><ref> to deduce that
        
    ^(σ)_p(μ, ν)^p
                   =^(√(σ^2-σ^2_0))_p(μ^σ_0, ν^σ_0)^p
       ≤ 6^p-1T2^T+1(M_q(μ^σ+ν^σ))^p/q( f_L^1^(σ_0)_1(μ, ν)/√(σ^2-σ^2_0))^1-p/q.

    

    
        Choosing σ_0=σ/√(2) in Corollary <ref><ref>,
        
    ^(σ)_p(μ, μ_n)^p
                ≤ 6^p-1T2^T+1 f^1-p/q_L^1(M_q(μ^σ+μ_n^σ))^1/r(^(σ/√(2))_1(μ, μ_n)/σ/√(2))^1/r'

        where r=q/p>1 and r' is the Hölder conjugate of r. Applying Hölder's inequality with 1/r+1/r'=1, 
        
    [^(σ)_p(μ, μ_n)^p]
                   ≤ C(M_q(μ^σ)+[M_q(μ_n^σ)])^1/r([^(σ/√(2))_1(μ, μ_n)]/σ/√(2))^1/r'
       ≤ C(M_q(μ)+σ^q M_q(ξ))^p/q([^(σ/√(2))_1(μ, μ_n)]/σ)^1/r'
       ≤ C(1+σ^p)([^(σ/√(2))_1(μ, μ_n)]/σ)^1/r'

        for some constant C>0 that depends only on d, T, p, q, M_q(μ). A close inspection of the proof of <cit.> shows that if q>dT+2, then for X∼μ,
        
    [_1^(σ/√(2))(μ, μ_n)]
                   ≤∫_(^d)^Tx[f_σ/√(2)∗μ(x)-f_σ/√(2)∗μ_n(x)]dx
       ≤(∫_(^d)^Tx^2/1+x^q dx)^1/2(∫_(^d)^T(1+x^q)[f^2_σ/√(2)(x-X)]/n dx)^1/2
       ≤ C(1+σ^q/2/σ^dT/2)1/√(n).

        for some C>0 that depends only on d, T, q, M_q(μ). We refer to the proof of <cit.> for computational details. This proves the desired result.
    



 §.§ Proofs for Section <ref>


In Lemma <ref>, we show that there is a Monge mapping from μ^σ to μ whose kernels are almost _p-optimal. We use this to prove Theorem <ref>. We then present proofs of Corollaries <ref> and <ref>.

    
        Let μ∈_p((^d)^T) and σ>0. For each >0 there exists a bicausal coupling π^∈(μ^σ, μ) given by π^=(𝕀, G^)_#μ^σ, where G^=(G^_1, …, G^_T):(^d)^T→ (^d)^T is a Borel measurable function, and π^ satisfies
        
    (∫x_1-y_1^p π^(dx_1, dy_1))^1/p≤ (1+)_p((μ^σ)_1, μ_1)

        as well as
        
    (∫x_t+1-y_t+1^p π^_x_1:t,y_1:t(dx_t+1, dy_t+1))^1/p≤ (1+)_p((μ^σ)_x_1:t, μ_y_1:t)

        for all t∈{1,2, …, T-1} and x_1:t, y_1:t∈ (^d)^t. Moreover, G^_t depends only on the first t coordinates, i.e. G^_t(x_1:T)=G^_t(x_1:t) for t∈{1,2, …, T-1}.
    
    
        Throughout the proof, >0 is fixed. Since (μ^σ)_1 is absolutely continuous, it can be approximated by Monge mappings. Precisely speaking, from <cit.>,
        
    _p((μ^σ)_1, μ_1)^p
                =inf_T: T_#(μ^σ)_1=μ_1∫x_1-T(x_1)^p μ^σ(dx_1).

        Hence there exists a map F^_1:^d→^d such that (F^_1)_#(μ^σ)_1=μ_1 and
        
    ∫x_1-F^_1(x_1)^p μ^σ(dx_1)≤ (1+)^p_p((μ^σ)_1, μ_1)^p.

        Let us define π^_1,1:=(𝕀, F^_1)_#(μ^σ)_1∈((μ^σ)_1, μ_1). For t∈{1,2, …, T-1}, let us denote by π^⋆_x_1:t, y_1:t∈((μ^σ)_x_1:t, μ_y_1:t) a _p-optimal coupling for each x_1:t, y_1:t∈ (^d)^t. It can be chosen in such a way that
        
    (^d)^t× (^d)^t ∋ (x_1:t, y_1:t)
                ↦π^⋆_x_1:t, y_1:t∈(^d×^d)

        is Borel measurable, where (^d ×^d) is endowed with the weak topology, see e.g. <cit.>. Thanks to Lemma <ref>, the map (<ref>) can be approximated by parameterized Monge couplings. Applying Lemma <ref> with 𝒳=^d and 𝒵=(^d)^t× (^d)^t, we find a measurable map F^_t+1:(^d)^t× (^d)^t×^d→^d such that
        
    ∫x_t+1-F^_t+1(x_1:t, y_1:t, x_t+1)^p (μ^σ)_x_1:t(dx_t+1)
            ≤ (1+)^p_p((μ^σ)_x_1:t, μ_y_1:t )^p.

        Set π^_x_1:t, y_1:t:=(𝕀, F^_t+1(x_1:t, y_1:t, · ))_#(μ^σ)_x_1:t∈((μ^σ)_x_1:t, μ_y_1:t) and define π^∈(μ^σ, μ) via
        
    π^(dx_1:T, dy_1:T)
                =π^_x_1:T-1, y_1:T-1(dx_T, dy_T)⋯π^_x_1, y_1(dx_2, dy_2)π^_1,1(dx_1, dy_1).

        Note that if X∼μ^σ, then Y∼μ, where Y is defined via the following recursive relation,
        
    Y_1=F^_1(X_1)  and  Y_t+1=F^_t+1(X_1:t, Y_1:t, X_t+1)  for  t∈{1,2, …, T-1}.

        Substituting (<ref>) iteratively, the construction of G^ is now straightforward.
    

    
        Let >0 and choose π^=(𝕀, G^)_#μ^σ∈(μ^σ, μ) as in Lemma <ref>. Note that
        
    _p(μ^σ, μ)
                ≤(∫∑_t=1^T x_t-y_t^p π^(dx,dy))^1/p≤∑_t=1^T(∫x_t-y_t^p π^(dx, dy))^1/p.

        Using the fact that _p((μ^σ)_1, μ_1)≤ (M_p(ξ_σ))^1/p=σ (M_p(ξ))^1/p, we deduce from Lemma <ref> that
        
    I^_1
                :=(∫x_1-y_1^p π^(dx, dy))^1/p≤ (1+)_p((μ^σ)_1, μ_1)
                ≤ (1+)σ(M_p(ξ))^1/p.

        Now, fix t∈{1,2, …, T-1}. Since G^_t depends only on the first t coordinates, we denote (G_1^(x), …, G_t^(x)) by G^(x_1:t). Also, let us denote the density of (ξ_σ)_1:t by
        
    (f_σ)_1:t(x_1:t)
                :=∫_(^d)^T-t f_σ(x_1:T)dx_t+1:T.

        Using the fact that (ξ_σ)_z_1:t has the density
        
    z_t+1↦(f_σ)_1:t+1(z_1:t+1)/(f_σ)_1:t(z_1:t),

        we compute that for measurable B⊆^d,
        
    (μ^σ)_x_1:t(B)
                =∫ (f_σ)_1:t(x_1:t-z_1:t) (μ_z_1:t∗ (ξ_σ)_x_1:t-z_1:t)(B) μ(dz_1:t)/(f_σ)_1:t∗μ_1:t(x_1:t).

        A similar identity was used in  <cit.>. From Lemma <ref> and convexity of _p^p applied to (<ref>), we obtain
        
    ∫x_t+1-y_t+1^p π^(dx, dy)
       = ∫x_t+1-y_t+1^p π^_x_1:t, y_1:t (dx_t+1, dy_t+1) π^(dx_1:t, dy_1:t)
       ≤∫  (1+)^p _p((μ^σ)_x_1:t, μ_y_1:t)^p π^(dx_1:t, dy_1:t)
       = (1+)^p∫_p((μ^σ)_x_1:t, μ_G^(x_1:t))^p μ^σ(dx_1:t)
       ≤ (1+)^p ∫∫ (f_σ)_1:t(x_1:t-z_1:t)_p(μ_z_1:t∗ (ξ_σ)_x_1:t-z_1:t, μ_G^(x_1:t))^pμ (dz_1:t)/(f_σ)_1:t∗μ_1:t(x_1:t)μ^σ(dx_1:t).

        Since x_1:t↦ (f_σ)_1:t∗μ_1:t(x_1:t) is the density of (μ^σ)_1:t, we use the change of variable w_1:t=x_1:t-z_1:t to obtain
        
    ∫∫ (f_σ)_1:t(x_1:t-z_1:t)_p(μ_z_1:t∗ (ξ_σ)_x_1:t-z_1:t, μ_G^(x_1:t))^pμ(dz_1:t)/(f_σ)_1:t∗μ_1:t(x_1:t)μ^σ(dx_1:t)
       =∫∫ (f_σ)_1:t(x_1:t-z_1:t)_p(μ_z_1:t∗ (ξ_σ)_x_1:t-z_1:t, μ_G^(x_1:t))^p μ(dz_1:t) dx_1:t
       =∫∫_p(μ_z_1:t∗ (ξ_σ)_w_1:t, μ_G^(z_1:t+w_1:t))^p (μ_1:t⊗ (ξ_σ)_1:t)(dz_1:t, dw_1:t)
       =[_p(μ_X_1:t∗ (ξ_σ)_σ Z_1:t, μ_G^(X_1:t+σ Z_1:t))^p],

        where (X, Z)∼μ⊗ξ. Using the above, we bound
        
    I^_t+1:=(∫x_t+1-y_t+1^p π^(dx, dy))^1/p
       ≤ (1+) [_p(μ_X_1:t∗ (ξ_σ)_σ Z_1:t, μ_G^(X_1:t+σ Z_1:t))^p]^1/p
       ≤ (1+) ([_p(μ_X_1:t∗ (ξ_σ)_σ Z_1:t, μ_X_1:t)^p]^1/p+ [_p(μ_X_1:t, μ_G^(X_1:t+σ Z_1:t))^p]^1/p)
       ≤ (1+)(σ (M_p(ξ))^1/p+[_p(μ_X_1:t, μ_G^(X_1:t+σ Z_1:t))^p]^1/p)
       ≤ (1+)(σ (M_p(ξ))^1/p+ω_μ^t,p(X_1:t-G^(X_1:t+σ Z_1:t)_L^p))
       ≤ (1+)(σ (M_p(ξ))^1/p+ω_μ^t,p(σ (M_p(ξ))^1/p+X_1:t+σ Z_1:t-G^(X_1:t+σ Z_1:t)_L^p))
       ≤ (1+)(σ (M_p(ξ))^1/p+ω_μ^t,p(σ (M_p(ξ))^1/p+∑_s=1^tI^_s)).

        Here, the third inequality is immediate from
        
    [_p(μ_X_1:t∗ (ξ_σ)_σ Z_1:t, μ_X_1:t)^p]
                   ≤[∫z_t+1^p (ξ_σ)_σ Z_1:t(dz_t+1)]
       =[∫z_t+1^p (f_σ)_1:t+1(σ Z_1:t, z_t+1)/(f_σ)_1:t(σ Z_1:t)dz_t+1]
       =∫∫z_t+1^p (f_σ)_1:t+1(z_1:t+1)/(f_σ)_1:t(z_1:t)dz_t+1(f_σ)_1:t(z_1:t)dz_1:t
       =σ^p M_p(ξ_t+1).

        
        
        
        
        
        The definition of ω_μ^t,p justifies the fourth inequality. The fifth inequality is the triangle inequality and monotonicity of ω_μ^t,p. To see the last inequality, note that (X+σ Z, G^(X+σ Z))∼π^ and
        
    X_1:t+σ Z_1:t-G^(X_1:t+σ Z_1:t)_L^p
                =(∫x_1:t-y_1:t^p π^(dx_1:t, dy_1:t))^1/p≤∑_s=1^t I^_s.

        Starting from the initial estimate (<ref>), we bound I^_t by h^t-1, p_μ using the recursive estimate (<ref>) along with Proposition <ref><ref> as follows,
        
    lim inf_→ 0I^_t≤ (1∨ (M_p(ξ))^1/p)(h^0,p_μ(σ)+th^t-1,p_μ(σ))

        for all t∈{2, 3,…, T}. This can be shown by the induction. When t=2, we compute from (<ref>) and (<ref>) that 
        
    lim inf_→ 0I^_2
                   ≤σ (M_p(ξ))^1/p+ω_μ^1,p(σ (M_p(ξ))^1/p+I^_1)
       ≤σ (M_p(ξ))^1/p+ω_μ^1,p(2σ (M_p(ξ))^1/p)
                ≤ (1∨ (M_p(ξ))^1/p)(σ+2ω^1,p_μ(σ)).

        Here, the last inequality follows from Proposition <ref><ref>. This proves (<ref>) when t=2. Assuming (<ref>) holds for I^_s, s∈{2,…, t}, a similar computation shows that
        
    lim inf_→ 0 I^_t+1   ≤σ(M_p(ξ))^1/p+ω^t,p_μ(σ(M_p(ξ))^1/p+I^_1+∑_s=2^tI^_s)
       ≤σ(M_p(ξ))^1/p
                +ω^t,p_μ(2σ(M_p(ξ))^1/p+∑_s=2^t(1∨ (M_p(ξ))^1/p)(h^0,p_μ(σ)+sh^s-1,p_μ(σ)))
       ≤σ(M_p(ξ))^1/p
                +(t+1)(1∨ (M_p(ξ))^1/p)ω^t,p_μ(∑_s=0^t-1h^s,p_μ(σ)).

        Hence (<ref>) holds for all t∈{2,3,…, T}.
        Plugging this back into (<ref>),
        
    _p(μ^σ, μ)
                ≤lim inf_→ 0∑_t=1^TI^_t≤σ(M_p(ξ))^1/p+∑_t=2^T(1∨ (M_p(ξ))^1/p)(h^0,p_μ(σ)+th^t-1,p_μ(σ)).

        This proves the desired estimate.
    


    
        <ref>  From Proposition <ref>, ω^t,p_μ(σ)≤ L σ^α if μ∈_p, α, L((^d)^T). Note from the construction of h^t,p_μ that h^1,p_μ(σ)=ω^1,p_μ(σ)≤ Lσ^α and
        
    h^2,p_μ(σ)
                =ω^2,p_μ(σ+h^1,p_μ(σ))
                ≤ L(σ+Lσ^α)^α≤ L(σ^α+L^ασ^α^2)
                ≤ (L∨ L^α+1)(σ^α+σ^α^2).

        It is straightforward from the mathematical induction that h^t,p_μ(σ)≤ C (σ^α+σ^α^2+…+σ^α^t) for some C>0 that depends only on L, α if t∈{1,2,…, T-1}. Applying this bound to Theorem <ref> yields the desired estimate.

        <ref>  From Theorem <ref>, we find
        
    sup_μ∈ K_p(μ^σ, μ)
                ≤ T(1∨ (M_p(ξ))^1/p)∑_t=0^T-1h^t,p_K(σ)

        where we define h^0,p_K(σ):=σ and h^t,p_K(σ):=sup_μ∈ Kω^t,p_μ(∑_s=0^t-1 h^s,p_K(σ)), t∈{1,2,…, T-1}. Note that if sup_μ∈ Kω^t,p_μ(δ)→ 0 as δ→ 0, the right hand side of (<ref>) goes to 0 as σ→ 0. The last statement is a straightforward application of a triangle inequality.
    

    
        From Corollaries <ref> and <ref><ref>, we estimate
        
    [_p(μ, (μ_n)^σ)^p]
                   ≤ C([^(σ)_p(μ, μ_n)^p]+[_p(μ^σ, μ)^p])
       ≤ C(1+σ^(q+p)/2/σ^(1+dT/2)(1-p/q))n^-(q-p)/(2q)
                +C∑_t=0^T-1σ^pα^t.

        We optimize the convergence rate of the right hand side by choosing σ = n^-γ where
        
    γ=1-p/q/(dT+2)(1-p/q)+2pα^T-1.

    




 §.§ Proofs for Section <ref>

      
          This is straightforward from the inequality
          
    _p(μ, ν)^p
                  ≤ 3^p-1^(σ)_p(μ, ν)
                  +3^p-1_p(μ, μ^σ)
                  +3^p-1_p(ν, ν^σ)

          applied to Theorem <ref> and Theorem <ref>.
      

      
          Choose ξ to be supported on {x∈ (^d)^T : x≤ 1}. From 0<σ<R, we have
          
    ∫_{x≥ 2R}x^p μ^σ(dx)
                     =∫_{x+σ z≥ 2R}x+σ z^p μ(dx)ξ(dz)
       ≤∫_{x≥ R}x+σ z^p μ(dx)ξ(dz)
                  ≤ 2^p-1∫_{x≥ R}x^p μ(dx)+2^p-1σ^p.

          Applying this bound to Proposition <ref> yields the desired result.
      
      
      
      <ref> By a translation argument we may assume that 0∈ F. Let μ, ν∈(F). Choose ξ to be supported on {x∈ (^d)^T : x≤ 1}. By combining the triangle inequality with Corollary <ref><ref> and Theorem <ref>, we compute
      
    _p(μ, ν)
                 ≤^(σ)_p(μ, ν)
              +2(_p(μ, μ^σ)∨_p(ν, ν^σ))
       ≤ C((1+σ)(_1(μ, ν)/σ)^1/p
              +∑_t=0^T-1h^t,p_μ(σ)∨ h^t,p_ν(σ))

      for some C>0 that depends only on d, T, (F). This proves the desired estimate.

      <ref> Let μ, ν∈(F)∩_p, α, L((^d)^T). Using Corollary <ref><ref> and Corollary <ref><ref>, we estimate
        
    _p(μ, ν)
                ≤ C( (1+σ)(_1(μ,ν)/σ)^1/p
                +∑_t=0^T-1σ^α^t)

        with some constant C>0 that depends only on d, T, L, α, (F). In (<ref>), choose σ=_1(μ, ν)^β for β=(pα^T-1+1)^-1 and we bound the right-hand side by
        
    (1+σ)(_1(μ,ν)/σ)^1/p
                +∑_t=0^T-1σ^α^t
       =(1+_1(μ, ν)^β)_1(μ,ν)^α^T-1/(pα^T-1+1)
                +∑_t=0^T-1_1(μ,ν)^α^t/(pα^T-1+1)
       ≤(1+(2(F))^β+∑_t=0^T-1(2(F))^(α^t-α^T-1)/(pα^T-1+1))_1(μ,ν)^α^T-1/(pα^T-1+1).

        Here, we make use of the fact that _1(μ, ν)≤ M_1(μ)+M_1(ν)≤ 2(F). This proves the desired result.
    



 §.§ Proofs for Section <ref>

In this section we prove Theorem <ref>, Theorem <ref> and Theorem <ref>. All the proofs are based on the following simple reformulation of the sequential convergence.


    Let  be a metric space. Suppose (x_n)_n∈⊆ and x∈. Then x_n→ x in  if and only if for every subsequence (x_n_k)_k∈ of (x_n)_n∈, there exists a subsequence of (x_n_k)_k∈ that converges to x.



    <ref> Suppose ^(σ_n)_p(μ_n, μ)→ 0. From Lemma <ref>, by passing to a subsequence, it is sufficient to assume that σ_n→σ_0 for some σ_0∈ [0,∞). From Proposition <ref>,
    
    ^(σ_0)_p(μ_n, μ)
            ≤_p^(σ_n)(μ_n, μ)
            +2σ_n-σ_0(M_p(ξ))^1/p≤_p^(σ_n)(μ_n, μ)
            +2σ_n-σ_0(M_p(ξ))^1/p.

    Since the right hand side converges to 0 as n→∞, we conclude that μ_n→μ in ^(σ_0)_p. We conclude that μ_n→μ in _p thanks to Proposition <ref>.


    <ref> Suppose now that _p(μ_n, μ)→ 0 and _1(μ_n, μ)/σ_n→ 0. From Lemma <ref>, we may assume that σ_n→σ_0 for some σ_0∈ [0,∞). We find from Theorem <ref> that for any R>0,
        
    lim sup_n→∞^(σ_n)_p(μ_n, μ)
                ≤ 6^p-1T2^T sup_n∈∫_{x≥ R}x^p (μ_n^σ_n+μ^σ_n)(dx).

        Note that μ^σ_n_n→μ^σ_0 and μ^σ_n→μ^σ_0 in _p. In particular, {μ^σ_n_n, μ^σ_n : n∈} is _p-relatively compact. Thus, the right hand side of (<ref>) can be made arbitrarily small as R→∞. This shows that ^(σ_n)_p(μ_n, μ)→ 0.



    <ref>  Suppose _p(μ_n, μ)→ 0. As in the proof of Theorem <ref>, it is sufficient to prove ^(σ_n)_p(μ_n, μ)→ when σ_n→σ_0 for some σ_0∈ [0,∞). If σ_0>0, then _1(μ_n, μ)/σ_n→ 0. By Theorem <ref><ref> this implies that ^(σ_n)_p(μ_n, μ)→ 0. Suppose now that σ_0=0. By the triangle inequality,
    
    ^(σ_n)_p(μ_n, μ)
            ≤_p(μ_n, μ)
            +2sup_ν∈ K_p(ν^σ_n, ν)

    where K={μ, μ_n :  n∈}. Note that K is _p-relatively compact. Thus, Proposition <ref> and Corollary <ref><ref> show that sup_ν∈ K_p(ν^σ_n, ν)→ 0. In conclusion, ^(σ_n)_p(μ_n, μ)→ 0.

    <ref> Suppose ^(σ_n)_p(μ_n, μ)→ 0 and h^t,p_μ_n(σ_n)→ 0 for all t∈{1,2,…, T-1}. Since ω^t,p_μ_n≤ h^t,p_μ_n, we have ω^t,p_μ_n(σ_n)→ 0. Using Lemma <ref>, we may assume that σ_n→σ_0 for some σ_0∈ [0,∞). If σ_0>0, fix >0 and choose N∈ such that sup_n≥ Nω^t,p_μ_n(σ_n)≤ and inf_n≥ Nσ_n≥σ_0/2. Note that
    
    lim sup_δ→ 0sup_n∈ω^t,p_μ_n(δ)
            ≤lim sup_δ→ 0∑_n=1^Nω^t,p_μ_n(δ)
            +sup_n≥ Nω^t,p_μ_n(σ_0/2)
            ≤ 0+sup_n≥ Nω^t,p_μ_n(σ_n)
            ≤.

    Since >0 was arbitrary and _p(μ_n, μ)→ 0, we deduce from Proposition <ref> that {μ_n :  n∈} is _p-relatively compact. Thus, _p(μ_n, μ)→ 0.

    Now, suppose σ_0=0. Then by the triangle inequality,
    
    lim sup_n→∞_p(μ_n, μ)
               ≤lim sup_n→∞(^(σ_n)_p(μ_n, μ)
            +_p(μ_n^σ_n, μ_n)
            +_p(μ^σ_n, μ))
       =lim sup_n→∞_p(μ_n^σ_n, μ_n).

    Here, the equality follows from Corollary <ref><ref> applied to K={μ}. It is easy to check from Theorem <ref> that h^t,p_μ_n(σ_n)→ 0 for all t∈{1,2,…, T-1} implies that _p(μ^σ_n_n, μ_n)→ 0. Hence, we conclude that _p(μ_n, μ)→ 0.






    From Proposition <ref> and the fact that ^(σ)_p≤ C^(σ)_p, it is obvious that if μ_n→μ in ^(σ)_p, then μ_n→μ in _p. Conversely, let us assume _p(μ_n, μ)→ 0. By choosing σ_n≡σ in Theorem <ref><ref>, we obtain that ^(σ)_p(μ_n, μ)→ 0. Therefore, ^(σ)_p metrizes the _p-topology. 














§ PROOF REGARDING EXAMPLE <REF>

    
        Let (_n)_n∈, (σ_n)_n∈⊆ (0, ∞) such that _n→ 0 and sup_n∈σ_n<∞. We define μ_n:=1/2δ_(_n, 1)+1/2δ_(-_n, -1), μ:=1/2δ_(0,1)+1/2δ_(0,-1), and ξ_σ_n:=(0, σ^2_n_2). Then we have
        
            
  *  _n/σ_n→ 0 if and only if _1(μ_n, μ)/σ_n→ 0 if and only if ^(σ_n)_p(μ_n, μ)→ 0.
            
  *  _n/σ_n→∞ if and only if ω^1,p_μ_n(σ_n)→ 0 if and only if _p(μ_n^σ_n, μ_n)→ 0.
        
    

    
        <ref> It is evident from _p(μ_n, μ)=_n that _n/σ_n→ 0 if and only if _1(μ_n, μ)/σ_n→ 0. The fact that _1(μ_n, μ)/σ_n→ 0 implies ^(σ_n)_p(μ_n, μ)→ 0 follows from Theorem <ref><ref>. Let us assume that ^(σ_n)_p(μ_n, μ)→ 0. We compute (μ^σ_n)_1=N(0, σ_n^2), (μ^σ_n)_x=1/2(1, σ_n^2)+1/2(-1, σ_n^2) and
        
    (μ^σ_n_n)_1=1/2(_n, σ_n^2)+1/2(-_n, σ_n^2),
       (μ^σ_n_n)_y
            =φ_σ_n(y-_n)/φ_σ_n(y+_n)+φ_σ_n(y-_n)(1,σ_n^2)+φ_σ_n(y+_n)/φ_σ_n(y+_n)+φ_σ_n(y-_n)(-1, σ^2_n)

        where φ(x):=(2π)^-1/2e^-x^2/2 and φ_σ_n(·):=(σ_n)^-1φ(·/σ_n). Since (μ^σ_n)_x does not depend on x, 
        
    ^(σ_n)_p(μ, μ_n)^p
               =inf_γ∈((μ^σ_n)_1, (μ^σ_n_n)_1)∫_^2x-y^p +_p((μ^σ_n)_x, (μ_n^σ_n)_y)^p γ(dx, dy)
       =_p((μ^σ_n)_1, (μ^σ_n_n)_1)^p
            +∫_-∞^∞_p((μ^σ_n)_x, (μ_n^σ_n)_y)^p(μ^σ_n_n)_1(dy).

    Using a change of variable, we write
    
    ∫_-∞^∞_p((μ^σ_n)_x, (μ_n^σ_n)_y)^p(μ^σ_n_n)_1(dy)
       =∫_-∞^∞_p((1, σ^2_n)/2+(-1,σ^2_n)/2, a_n(y)(1, σ^2_n)+b_n(y)(-1,σ^2_n))^pφ(y)dy,

    where a_n(y):=φ(y)/(φ(y)+φ(y+2_n/σ_n)) and b_n(y):=1-a_n(y). If _n/σ_n↛ 0, then up to a subsequence, we may assume that _n/σ_n→ c for some c∈ (0, ∞]. From (<ref>) and the Fatou lemma,
    
    0=lim_n→∞_p^(σ_n)(μ, μ_n)^p
               ≥lim inf_n→∞∫_-∞^∞_p((μ^σ_n)_x, (μ_n^σ_n)_y)^p(μ^σ_n_n)_1(dy)
       = lim inf_n→∞∫_-∞^∞1/2 (_p((μ^σ_n)_x, (μ_n^σ_n)_ϵ_n+σ_n y)^p + _p((μ^σ_n)_x, (μ_n^σ_n)_-ϵ_n+σ_n y )^p ) φ(y)dy
       ≥∫_-∞^∞_p(δ_1+δ_-1/2, φ(y)δ_1+φ(y+2c)δ_-1/φ(y)+φ(y+2c))^pφ(y)dy
            >0.

    Thus, ^(σ_n)_p(μ, μ_n)→ 0 only if _n/σ_n→ 0. 
    
    <ref> It is easy check that ω^1,p_μ_n(σ_n)=(σ_n/_n)∧ 2. Hence _n/σ_n→∞ if and only if ω^1,p_μ_n(σ_n)→ 0. From Theorem <ref>, we have _p(μ^σ_n_n, μ_n)
        ≤ C (σ_n+ω^1,p_μ_n(σ_n)). Hence ω^1,p_μ_n(σ_n)→ 0 yields _p(μ^σ_n_n, μ_n)→ 0. Now, let us assume that _p(μ^σ_n_n, μ_n)→ 0. Note that
    
    _p(μ_n^σ_n, μ_n)^p
            =inf_γ∈((μ_n^σ_n)_1, (μ_n)_1)∫_^2x-y^p +_p((μ_n^σ_n)_x, (μ_n)_y)^pγ(dx, dy).

    Choose γ^⋆ to be optimal for (<ref>). Let Z∼(0,1), c_n(x):=φ_σ_n(x-_n)/(φ_σ_n(x-_n)+φ_σ_n(x+_n)) and d_n(x):=1-c_n(x). Then
    
    _p(μ_n^σ_n, μ_n)^p
               ≥∫_{y=_n}_p((μ_n^σ_n)_x, δ_1)^pγ^⋆(dx, dy)
            +∫_{y=-_n}_p((μ_n^σ_n)_x, δ_-1)^pγ^⋆(dx, dy)
       =∫_{y=_n} c_n(x)[σ_n Z^p]+d_n(x)[2+σ_n Z^p] γ^⋆(dx, dy)
       +∫_{y=-_n} c_n(x)[2+σ_n Z^p]+d_n(x)[σ_n Z^p] γ^⋆(dx, dy)
       ≥ ([σ_n Z^p]+[2+σ_n Z^p])∫_-∞^∞ (c_n(x)∧ d_n(x))(μ^σ_n_n)_1(dx).

    We compute
    
    ∫_-∞^∞ (c_n(x)∧ d_n(x))(μ^σ_n_n)_1(dx)
               =1/2∫_-∞^∞ (φ_σ_n(x-_n)∧φ_σ_n(x+_n))dx
       =1/2(∫_-∞^-_n/σ_n+∫__n/σ_n^∞)φ(x)dx.

    Hence _p(μ^σ_n_n, μ_n)→ 0 implies σ_n/_n→∞.
    


    



§ MONGE MAPS


    Let 𝒜, ℬ, 𝒵 be Polish spaces. We say that π is a kernel from 𝒵 to 𝒜 if π is a Borel measurable map from 𝒵 to (𝒜). Here, (𝒜) is endowed with the weak topology. We will denote the probability measure π(z) by π^z. We use a similar notation for functions. Given a Borel measurable function T:𝒵×𝒜→ℬ, we define T^z:𝒜→ℬ via T^z(a)=T(z,a). For a partition ℛ of 𝒳, we define its mesh via ℛ=sup_A∈ℛ(A).

    
        Let 𝒳,𝒵 be Polish spaces and let π be a kernel from 𝒵 to 𝒳×𝒳. Let μ^z and ν^z denote the pushforward measure of π^z through a projection on to the first coordinate and the second coordinate, respectively, i.e. π^z∈(μ^z, ν^z). Suppose μ^z, ν^z∈_p(𝒳) and μ^z has no atoms for all z∈𝒵. Then there exist Borel measurable maps T_n:𝒵×𝒳→𝒳 such that (𝕀, T^z_n)_#μ^z∈(μ^z, ν^z) and
        
    ∫_𝒳×𝒳 d_𝒳(x,y)^p π^z(dx, dy)
                =lim_n→∞∫_𝒳 d_𝒳(x, T_n^z(x))^p μ^z(dx).

        Here d_𝒳 is a metric on 𝒳 that induces the Polish topology.
    
    
        It follows from <cit.> that there exits a Borel measurable map T:𝒵×𝒳× [0,1]→𝒳× [0,1] such that
        
            
  * T^z:=T(z, ·):𝒳× [0,1]→𝒳× [0,1] is a Borel isomorphism, i.e. T^z is Borel measurable, bijective and its inverse is also Borel measurable,
            
  * for the Lebesgue measure λ on [0,1], (𝕀, T^z)_#(μ^z⊗λ)∈(μ^z⊗λ, ν^z⊗λ) and its projection on 𝒳×𝒳 is π^z.
        
        Now let us choose a partition ℛ_n of 𝒳 that consists of at most countable sets and satisfies lim_n→∞ℛ_n=0. We apply <cit.> to find Borel measurable maps Φ_n:𝒵×𝒳→𝒳× [0,1] such that
        
        
            
  * Φ_n^z:=Φ_n(z, ·):𝒳→𝒳× [0,1] is Borel isomorphism,
            
  * (Φ^z_n)_#(μ^z|_A)=(μ^z|_A)⊗λ for all A∈ℛ_n where μ^z|_A(·):=μ^z( ·∩ A).
        
        Let P_𝒳 and P_𝒵 be projection maps on 𝒳 and 𝒵 respectively and define a map T_n:𝒵×𝒳→𝒳 via T_n = P_𝒳∘ T ∘ (P_𝒵, Φ_n). Note that π^z_n:=(𝕀, T^z_n)_#μ^z∈(μ^z, ν^z) by (b) and (d). Similarly as in the proof of <cit.>, we can show that
        
    π^z_n(A× B)
                =π^z(A× B)  for any  A, B∈ℛ_n  and  z∈𝒵.

        Indeed, for A, B∈ℛ_n,
        
    π^z_n(A× B)
                =μ^z(A∩ (T^z_n)^-1(B))
                   =μ^z|_A((Φ^z_n)^-1∘ (T^z)^-1∘ (P_𝒳)^-1(B))
       =μ^z|_A⊗λ ((T^z)^-1∘ (P_𝒳)^-1(B))
       =μ^z⊗λ ((A× [0,1])∩ (T^z)^-1(B× [0,1]))
       =(𝕀, T^z)_#(μ^z⊗λ)((A× [0,1])× (B×[0,1]))
       =π^z(A× B).

        The first equality comes from the definition of π_n^z. For the second equality, we use the definition of T^z_n. The property (d) implies the third equality. The fourth and the fifth equality are straightforward from the pushforward operator and the last equality is obtained from the property (b). Thus, we deduce from <cit.> that π^z_n →π^z in _p on (𝒳×𝒳)^2. This implies the desired results.
    





abbrv





 Paper Title: Shape-restricted transfer learning analysis for generalized linear regression model.






Shape-restricted transfer learning analysis for generalized linear regression models
    Pengfei Li^1, Tao Yu^2, Chixiang Chen^3,  and Jing Qin^4
    
====================================================================================






^1Pengfei Li is Professor,
Department of Statistics and Actuarial Sciences,
University of Waterloo,
Waterloo, ON, Canada, N2L 3G1

(Email: pengfei.li@uwaterloo.ca)

^2Tao Yu is Associate Professor,
Department of Statistics and Data Science,
National University of Singapore, Singapore, 117546

(Email: yu.tao@nus.edu.sg)


^3Chixiang Chen is 
Assistant professor, Department of Epidemiology and Public Health,
University of Maryland School of Medicine, MD 21201, U.S.A.

 (Email: Chixiang.Chen@som.umaryland.edu)

^4Jing Qin is Mathematical Statistician, National Institute of Allergy and Infectious Diseases, National Institutes of Health, MD 20892, U.S.A.

 (Email: jingqin@niaid.nih.gov)





§ ABSTRACT 

Transfer learning has emerged as a highly sought-after and actively pursued research area within the statistical community. The core concept of transfer learning involves leveraging insights and information from auxiliary datasets to enhance the analysis of the primary dataset of interest.
In this paper, our focus is on datasets originating from distinct yet interconnected distributions. We assume that the training data conforms to a standard generalized linear model, while the testing data exhibit a connection to the training data based on a prior probability shift assumption. Ultimately, we discover that the two-sample conditional means are interrelated through an unknown, nondecreasing function. We integrate the power of generalized estimating equations with the shape-restricted score function, creating a robust framework for improved inference regarding the underlying parameters.
We theoretically establish the asymptotic properties of our estimator and demonstrate, through simulation studies, that our method yields more accurate parameter estimates compared to those based solely on the testing or training data. Finally, we apply our method to a real-world example. 











Keywords: Estimating equation; Generalized linear model; Prior Probability shift; Shape-restricted inference; Transfer learning



§ INTRODUCTION
 

In the era of big data and diverse sources of information, statistical transfer learning has emerged as a powerful paradigm to address challenges posed by data heterogeneity and limited sample sizes. Traditional statistical methods often assume that the training and testing data are drawn from the same distribution <cit.>, an assumption that may not hold true in many real-world scenarios. 
The motivation behind statistical transfer learning stems from the recognition that data collected from one domain can contain valuable information benefiting the learning process in another domain <cit.>. This is particularly relevant when the target domain has limited labeled data or exhibits different statistical properties compared to the source domain. By harnessing the wealth of knowledge from related domains, statistical transfer learning offers a way to enhance predictive accuracy, generalization, and interpretability in various applications, such as the development of precision medicine  <cit.>, language recognition <cit.>, medical diagnosis <cit.>, and recommender systems <cit.>.

The predictive accuracy of machine learning models is significantly impacted by the presence of data shift, which occurs when the distribution of the data used for model training differs substantially from the distribution of the data encountered during deployment. This issue is particularly pronounced in clinical AI systems, where machine learning algorithms play a crucial role in discerning patterns from clinical data. Dataset shift can lead to severe consequences, adversely affecting model performance and patient outcomes in such systems.
The impact of dataset shift becomes notably apparent in the case of the sepsis-alerting model at the University of Michigan Hospital <cit.>. 
Developed by Epic Systems, this model effectively identified potential sepsis cases using historical patient data.
However, during the coronavirus disease 2019 pandemic, the model's performance deteriorated due to shifts in patients' demographic characteristics and other relevant factors.
This dataset shift fundamentally altered the relationship between fevers and bacterial sepsis, resulting in spurious alerting. As a consequence, the hospital's clinical AI governing committee deactivated the model for patient safety reasons.
Therefore, it is crucial to explore methods that can appropriately combine data information from similar studies when conducting statistical learning across various scientific fields, including economics, machine learning, distributed statistical inferences, and others.



Data shift can manifest in various forms. One extensively studied transfer learning problem involves binary classification within the framework of domain adaptation. In this scenario, the distributions of data pairs 
(X, Y) are assumed to be drawn from related but distinct distributions defined on ^p×{0,1}. 
For simplicity, we focus on 
two distributions–namely, the training and testing distributions. In this framework, two general strategies are employed. The first strategy involves making no additional assumptions about the relationship between distributions but instead suggesting a measurement of the divergence of the distributions. This measurement is then used to establish and quantify the effectiveness of the classification rules  <cit.>. The second strategy involves imposing structural assumptions between the training and testing distributions. Common assumptions include covariate shift and label shift (or more generally, prior probability shift). 
The covariate shift refers to the change in the distribution of the covariates X in the training and testing data <cit.>. Label shift assumes that the distributions of the covariates X may remain similar, while the distributions of Y|X can be significantly different <cit.>. Under this assumption, <cit.> considered the linear regression model, i.e., Y = X^Tβ + ϵ, when Y is a continuous variable. In their approach, they proposed to estimate β by a weighted sum of the corresponding least square estimates of β from the training and testing data. By assuming that the regression coefficients are different, but model structures are the same, for training and testing data <cit.> and <cit.> considered the transfer learning method for high dimensional linear models and generalized linear models with a Lasso penalty. 

The transfer learning method for the generalized linear model in the context of “prior probability shift" has not been thoroughly explored in the literature. This paper aims to introduce a method under this setup that can effectively address the heterogeneity between the training and testing data. 
Specifically, we assume that the distributional difference between the training data and the testing data lies in the conditional distributions of Y|X, which is captured by the conditional density relationship:

    f^*(y|x) ∝π(y) f(y|x),

where π(y) is an unknown selection function, and f(y|x) and f^*(y|x) are the conditional densities of Y|X = x for the training and testing data, respectively.
In general, there is no strict requirement for the marginal densities of X in the training and testing data to be identical. Consequently, the conditional densities f(x | y) in the training and testing data might show variations. Essentially, our assumption is somewhat less stringent than the previously mentioned probability shift assumption.


We observe that Assumption (<ref>) is well-motivated in practical scenarios. Let D=1 or 0 denote whether an individual data point is observed or not. If

    P(D=1|X = x,Y=y)=P(D=1|Y=y)=π(y)

depending on the outcome Y only, then, by Bayes's formula, we can show that

    Y|{X = x, D=1}∼π(y)f(y|x)/∫π(y)f(y|x)dy.












In other words, if the selection bias function depends solely on the outcome, it leads to a scenario resembling a prior probability shift issue. 
It's worth noting that model (<ref>) demonstrates a high degree of flexibility. It accommodates situations where the testing data stem from a selection-biased variant of the training data. Furthermore, it allows for differences in covariate densities between the training and testing data. In practical applications, such as clinical trials conducted across various research institutes, variations in patient inclusion criteria can result in distinct covariate distributions.









The issue known as the prior probability shift problem transforms into what is commonly referred to as the label shift problem when the outcome Y is discrete. For example, when Y represents an individual's reported disease status and X encompasses symptoms and medical test outcomes obtained from standard equipment, this challenge is characterized as the label shift problem.
In some instances, it may be reasonable to assume that the distribution of X is consistent across various studies. However, given a specific set of observations X = x, the reported disease status from these studies may be influenced by Y. This influence can be attributed to various factors, such as differences in the knowledge and education backgrounds of doctors or environmental impacts. 
The aforementioned sepsis-alerting model <cit.> developed by the University of Michigan Hospital stands as a concrete example of this label shift phenomenon.


We assume that Equation (<ref>) holds, and that observations in the training data follow the exponential family with 
E(Y|X = x)=μ(x^Tβ). 
Under these assumptions, 
 we demonstrate  (refer to  Section <ref>) that for the testing data,

    E(Y|X = x) = ψ(μ(x^T β)),

with ψ(·) being a monotone function. This is inline with the development in <cit.>, in which they assumed that the regression models for binary outcomes Y in both the training and testing data are linked through a known nondecreasing link function and a low-dimensional transformation of the covariates. This innovative perspective draws inspiration from the Cox proportional hazard model.
In our current paper, we deviate from the assumption of a binary outcome and, instead, focus on more general regression functions in both the training and testing data. These functions are interconnected through an unknown monotone transformation, presenting an intriguing challenge. Essentially, this shift transforms our problem into a shape-restricted transfer learning scenario, paving the way for exciting avenues of exploration and investigation. Notably, shape-restricted inference has found applications in various domains, including AI <cit.>. In transfer learning scenarios, the incorporation of shape restrictions, such as monotonicity, proves particularly advantageous, as it enables us to align with prior knowledge and enforce domain-specific constraints.


It's worth noting that assuming a generalized linear model for the training data, Equation (<ref>) implies Equation (<ref>). However, Equation (<ref>) presents a more general condition than Equation (<ref>).
In particular, when the outcome variable Y is binary, <cit.> and <cit.> have made similar assumptions concerning a monotone link function for the transfer learning problem. Their primary focus has been on achieving minimax optimal rates of convergence in their respective studies.
This paper focuses on the assumption stated in Equation (<ref>).

The rest of the paper is organized as follows.  In Section <ref>, we present the model assumptions for both the training and testing data, and introduce our estimating equation-based method for β. This method incorporates information from both the training and testing data, assuming that ψ(·) in (<ref>) is unknown.
In Section <ref>, we theoretically establish the asymptotic normality of our β estimates. Section <ref> presents simulation studies, while Section <ref> applies our method to a real data example. Finally, Section <ref> concludes the paper with a discussion. For ease of presentation, technical details are provided in the Appendix and supplementary material.





§ ESTIMATION METHOD
 

Consider a set of training data: (Y_1,i, X_1,i), i=1, …, n, and a set of testing data: (Y_2,j, X_2,j), j=1, …, m.
Assume that the training data consists of independent and identically distributed (i.i.d.) copies of (Y_1, X_1), which follows a standard parametric generalized linear model <cit.>. Specifically, assume that the conditional density of Y_1 given the covariate X_1 = x belongs to a canonical exponential family:

    f(y|x) = exp{y θ(x) - b(θ(x))/a(ϕ)  +  c(y, ϕ)}.

Here, a(·), b(·), and c(·, ·)
are known functions,  θ(·) is the canonical parameter, and ϕ is the dispersion parameter. 
In the context of the parametric generalized linear model, we have E(Y_1|X_1=x) = μ_1(x^T β), where μ_1(·) is a known increasing function. 
Subsequently, the following relationships hold:

    μ_1(x^T β)    =    E(Y_1|X_1=x) = b'(θ(x)), 
    var(Y_1|X_1=x)    =    a(ϕ) b”(θ(x)) ≡ a(ϕ) · V(x^T β),

where V(·) is the variance function with a known function form. 
It is straightforward to check 

    ∂log f(y|x)/∂β = μ_1'(x^T β) /a(ϕ) V(x^T β)x{ y - μ_1(x^T β)}.

Assuming that the dispersion parameter ϕ is a constant across the observations, the maximum likelihood estimate for β can be obtain by solving the equations ψ_1,n(β) = 0, where 

    ψ_1,n(β)    =   1/n∑_i=1^n μ_1'(X_1,i^Tβ)/V(X_1,i^Tβ) X_1,i{ Y_1,i - μ_1(X_1,i^Tβ) }.


For the testing data (Y_2,j, X_2,j), j=1,…,m, we assume that they are i.i.d. copies of (Y_2, X_2). 
The conditional distribution of Y_2 is assumed to be related to that of the training data. Specifically, we assume that the conditional density of Y_2|X_2 = x, denoted by f^*(y|x), satisfies (<ref>), with π(y) representing an unknown selection function. 
Consequently,

    f^*(y|x) = π(y) f(y|x)/∫π(y) f(y|x) dy.

The following proposition holds. 
 
Assume the conditional density of Y_2|X_2 = x in (<ref>), with f(y|x) given by (<ref>). Then, there exists a nondecreasing function μ_2(·) such that

    E(Y_2|X_2 = x) = μ_2(x^T β).


Combining (<ref>) and (<ref>), we obtain 

    f^*(y|x)
        =   exp[ y θ(x)/a(ϕ)+{logπ(y)+c(y,ϕ)}]/∫exp[ y θ(x)/a(ϕ)+{logπ(y)+c(y,ϕ)}] dy
       =   exp[ y θ(x)-  b_π(θ(x), ϕ)/a(ϕ)+{logπ(y)+c(y,ϕ)}],

where 

    b_π(θ(x), ϕ) = a(ϕ) log∫exp[ y θ(x)/a(ϕ)+{logπ(y)+c(y,ϕ)}] dy.

Denote by E_Y_2|X_2 = x(·) the expectation with respect to Y_2 under the density f^*(y|x). 
Due to the fact E_Y_2|X_2 = x[ ∂log f^*(Y_2|x)/∂{θ(x) }] = 0,
we have

    E_Y_2|X_2 = x{Y_2 - ∂ b_π(θ(x), ϕ)/∂θ(x)} = 0,

or equivalently,

    E(Y_2|X_2 = x) = ∂ b_π(θ(x), ϕ)/∂θ(x).

Furthermore, based on the fact that 

    E_Y_2|X_2 = x[ ∂^2 log f^*(Y_2|x)/∂{θ(x) }^2] + E_Y_2|X_2 = x([ ∂log f^*(Y_2|x)/∂{θ(x) }]^2 ) = 0,

we have

    ∂^2 b_π(θ(x), ϕ)/∂{θ(x)}^2   =   1/a(ϕ)· E_Y_2|X_2 = x[{Y_2 - ∂ b_π(θ(x), ϕ)/∂θ(x)}^2] 
       =   var(Y_2|X_2 = x)/a(ϕ)≥ 0,

where “=" holds if and only if Y_2 = E(Y_2|X_2 = x) almost surely. Combining (<ref>) and (<ref>), we conclude that E(Y_2|X_2 = x) is a nondecreasing function of θ(x), and therefore is a nondecreasing function of x^T β.  We complete the proof of the proposition. 

With Proposition <ref>, the testing data can be analyzed using the monotone single index model <cit.>. Specifically, we can employ the following profiling procedure to establish a set of estimating functions based on the testing data. For each given β, we estimate μ_2(·) by

    μ_2,β(·) = min_μ_2∑_j=1^m {Y_2,j - μ_2(X_2,j^Tβ)}^2,

subject to μ_2(·) being a nondecreasing function. 
This can be solved by the standard PAVA algorithm <cit.>. Employing similar ideas as <cit.> and <cit.>, we can construct a set of estimating functions:

    ψ_2,m(β) = 1/m∑_j=1^m X_2,j{Y_2,j-μ_2,β(X_2,j^Tβ)}.

Based only on the testing data, a β estimate can be obtained by solving the equation ψ_2,m(β) = 0. 

Combining (<ref>) with (<ref>), let 

    ψ_n,m(β) = λ_n,mψ_1,n(β) + (1-λ_n,m) ψ_2,m(β),

where λ_n,m = n/(n+m). We hope to define β to be the root of ψ_n,m(β) = 0. However, ψ_n,m(·) may not be a continuous mapping, and its root may not exist. Similar to <cit.>, we define β to be the zero crossing of ψ_n,m(β). The zero crossing of a function (or a mapping) is defined below.


For a function ψ: →, x is called the zero-crossing of ψ if every open neighborhood of x contains x_1, x_2 such that ψ(x_1)ψ(x_2) ≤ 0.
For a mapping ψ: →^d for d≥ 2, x is called the zero-crossing of ψ if x is the zero-crossing of each component of ψ.


In most practical scenarios, it is reasonable to assume that the size of the training data, i.e., n, is less than or comparable to that of the testing data, i.e., m; in the sense λ_n,m→λ > 0. Some discussion on this assumption can be found in <cit.> and <cit.>. However, from both a methodological and theoretical standpoint, it is possible for n ≪ m or, more rigorously, for n/m → 0, resulting in λ_n,m→λ = 0. Intuitively, when λ_n,m is small, in other words, when we have much more testing data than training data, the contribution of the training data to the β estimation (through ψ_1,n(β)) is limited; therefore, β relies mainly on the testing data (through ψ_2,m(β)).
On the other hand, with only the testing data, model (<ref>) is identifiable only when a constraint is imposed on β, such as
β_2 = 1. Theoretically, the zero crossing of ψ_2,m(β) only exists asymptotically for p-1 out of its p components, such as the 2nd to pth component if β_0,1≠ 0, where β_0 denotes the true value of β, and β_0,1 denotes the first component of β_0. See Section <ref> and the corresponding technical developments in the supplementary material for details.
Practically, when λ_n,m is very small, we suggest estimating β as the zero crossing of  ψ_n,m(β)_2:p under the constraint β_2 = 1, where ψ_n,m(β)_2:p is a vector composed of the 2nd to pth component of ψ_n,m(β). 




§ ASYMPTOTIC PROPERTIES
 

In this section, we will establish the asymptotic properties of β. Besides the notations in Section <ref>, we need some additional notations. Let β_0 and μ_2,0(·) denote the true values of β and μ_2(·), respectively. Also, let φ_β(t) = E(X_2|X_2^Tβ = t), μ_2, β(t) = E(Y_2|X_2^Tβ = t); thus μ_2,0(·) = μ_2,β_0(t).  
Denote

    A(β_0)  
       =   λ E[ {μ_1'(X_1^Tβ_0)}^2/V(X_1^T β_0) X_1 X_1^T ] 
           + (1-λ)E[ {X_2 - φ_β_0(X_2^T β_0)}{X_2 - φ_β_0(X_2^T β_0)}^T μ_2,0'(X_2^Tβ_0 )   ], 
    Σ_1    =   var[ μ_1'(X_1^Tβ_0)/V(X_1^Tβ_0) X_1 { Y_1 - μ_1(X_1^T β_0) }],
    Σ_2    =   var[ {X_2 - φ_β_0(X_2^T β_0 ) }{Y_2 - μ_2,0(X_2^Tβ_0)}],

where μ'(t) = ∂μ(t)/∂ t for μ(t) being a function differentiable at t. Furthermore, let 

    lim_n→∞, m →∞λ_n,m = λ.

As discussed earlier, we need to consider two possibilities: λ > 0 and λ = 0. Let us first consider the case where λ >0. Note that this includes λ = 1 as a special case, which implies that we have much more training data than testing data.

We impose the following regularity conditions in the development of the theoretical results. They are not necessarily the weakest possible. 


 
β∈, F_X_1(x) and F_X_2(x) are supported on , where both  and  are compact subspaces of ^p, and F_X_s(·) for s=1 or 2 denotes the cumulative distribution function (c.d.f.) of X_s.  


 
There exists a universal constant C>0, such that for any β_1, β_2 ∈, 

    sup_t∈∫| I(β_1^T x≤ t) - I(β_2^T x≤ t )  | d F_X_2(x) ≤ C β_1 - β_2_1,

where ·_q denotes the l_q norm. 


 
The conditional probability density function f_X_2|X_2^Tβ = t (x) is continuously differentiable in β and t. 


 






μ_2,β(t) is continuously differentiable for both β∈ and t∈{x^Tβ: x∈, β∈}, and there exists a universal constant η_0 >0, such that 

    inf_t∈{x^Tβ: x∈, β∈}, β - β_0_2 ≤η_0∂μ_2,β(t)/∂ t> 0.



 
The density function of X_1, denoted by f_X_1(x), is continuously differentiable for x∈, and E(X_1X_1^T) is of full rank.  


 
For s=1 or 2, and any integer k≥ 2, E(|Y_s|^k|X_s) ≤ k! M^k-2 v/2 for some universal constants M and v.


 
For t ∈{x^Tβ: x∈, β∈}, μ_1(t) is nondecreasing and continuously differentiable, μ_1'(t) is bounded away from zero and infinity; and V(t) is continuously differentiable and is bounded away from 0 and ∞.  


 
For any β_1, β_2∈, such that β_1 ≠ c β_2, for all c ∈, and c≠ 0, we have, 

    E{var( β_1^T X_2 | β_2^T X_2 )} >0.

 

 
Note that Condition <ref> implies that if β_1 ≠ c β_2, for all c ∈, and c≠ 0, then,

    P(β_1^T X_2 ≠  E( β_1^T X_2 | β_2^T X_2 ) ) >0;

since otherwise, β_1^T X_2 = E( β_1^T X_2 | β_2^T X_2 ) almost surely, which implies 

    var( β_1^T X_2 | β_2^T X_2 ) =  E[{β_1^T X_2 -  E( β_1^T X_2 | β_2^T X_2 )}^2 | β_2^T X_2 ] = 0,

almost surely, which contradicts Condition <ref>. 





Theorem <ref> below establishes the existency and asymptotic properties of β when λ >0. 

 
Assume Conditions <ref>–<ref>. 
Denote by β the zero crossing of ψ_n,m(β) (if it exists).
We have the following:


  (1) When n→∞ and m →∞, a zero crossing β of ψ_n,m(β) exists with probability tending to 1. 

  (2) If β is a zero crossing of ψ_n,m(β), we have β→β_0 in probability.

  (3) For β being a zero crossing of ψ_n,m(β), we have

    √(n+m)( β- β_0 ) →  A_0(β_0)^-1· N(0, λΣ_1 + (1-λ) Σ_2   ),

in distribution. 



Next, we consider that λ = 0. In this case, we shall define β to be the zero crossing of ψ_n,m(β)_2:p under the constraint β_2 = 1; here for a vector x, x_i:j denotes the vector formed of the ith to jth entries of x. Theorem <ref> below establishes the asymptotic properties of this β. We need the following notation.

    A(β_0)  
       =    E[ {X_2 - φ_β_0(X_2^T β_0)}_2:p{X_2 - φ_β_0(X_2^T β_0)}_2:p^T μ_2, 0'(X_2^Tβ_0 )   ] 
          ×(β_0,2:pβ_0,2:p^T + I_p-1),
    Σ_2    =   var[ {(X_2)_2:p - φ_β_0(X_2^T β_0 )_2:p}{Y_2 - μ_2,0(X_2^Tβ_0)}].





 
Assume Conditions <ref>–<ref>. Suppose λ = 0 and β_0,1≠ 0. 
Denote by β the zero crossing of ψ_n,m(β)_2:p (if it exists) under the constraint β_2 = 1.
We have the following:


  (1)  There exists a universal constant c>0, such that when n→∞ and m →∞, a zero crossing β of ψ_n,m(β)_2:p with β_1 > c exists with probability tending to 1. 

  (2) Assume that β is a zero crossing of ψ_n,m(β)_2:p such that β_1 > c for a universal constant c>0, we have β→β_0 in probability.

  (3) For β given in Part (2), we have

    √(n+m)(β_2:p - β_0, 2:p)→A(β_0)^-1 N(0, Σ_2),

in distribution.




For presentational continuity, we sketch the proof for Part (3) of Theorems <ref> and <ref> in the Appendix, and relegate the lengthy technical details to the supplementary material. 


 


§ SIMULATION STUDY
 


 §.§ Data Simulation

We consider the following simulation setups. For both the training and testing data, we generate the covariate X_1,i and X_2,j for all i and j to be i.i.d. copies of X = (U_1, U_2, U_3)^T, where

    U_1∼Unif[-1.5,1.5],   U_2∼Bernoulli(0.5),    U_3∼ N(0,1),

where U_1, U_2, and U_3 are independent. Given X = x≡ (u_1, u_2, u_3)^T, the responses of the training and testing data are then simulated based on two models. 


  * Model 1 (Gamma model)

The training data are simulated from f(y|x), which is the Gamma distribution with mean exp(u_1 + u_2 + u_3) and shape parameter equal to 2. The testing data are simulated from f^*(y|x) ∝π(y) f(y|x) with 

    π(y)=exp(-0.5y)/1+exp(-0.5y).



  * Model 2 (Normal model)

The training data are simulated from

    Y=2+u_1+u_2+u_3+ϵ,   ϵ∼ N(0,1).

Therefore, f(y|x) is the density function of N(2+u_1 + u_2+u_3, 1). 
The testing data are simulated from f^*(y|x) ∝π(y) f(y|x) with 

    π(y)=exp(-0.5y)/1+exp(-0.5y).



Thus, for both models, we have β_0 = (1,1,1)^T. We examine three combinations of sample sizes:  (750,250), (500,500), and (250,750). For each model configuration, we perform 1000 repetitions. 



 §.§ Results


To illustrate the advantages of our method, which combines training and testing data to improve β estimation, we compare it to two other approaches: using only training data and using only testing data. Specifically, we consider 


    
  * the maximum likelihood estimate based only on the training data, called MLE estimate;
    
  * the estimating equation estimate based only on the testing data, called EE estimate;
    
  * the estimate based on the estimating equations (<ref>), which combine the training data and the testing data, called our estimate. 


To obtain the EE estimate, we impose the constraint β_2 = 1. Therefore, to compare the performance of the methods, we standardise the estimates by

    η_i=β_i/√(β_1^2+β_2^2+β_3^2),      i=1,2,3.

The true value of  (η_1,η_2,η_3) is given by  (1/√(3),1/√(3),1/√(3)). 

We use the relative bias (RB) and mean square error (MSE) as the comparison criteria. For a true parameter η_0 and the corresponding estimates η_1, …η_N based on N repetitions, its RB in percentage and MSE are defined to be

    RB = ∑_i=1^N (η_i - η_0 )/N× 100,   MSE = ∑_i=1^N (η_i - η_0 )^2 /N.


Table <ref> presents the RBs and MSEs of the parameter estimates obtained over 1000 repetitions. Our method, which combines information from both the training and testing data, consistently produces the smallest MSEs for all model configurations. When the sample size of the training data is much larger than that of the testing data, specifically with n=750, m=250, and λ_n,m = 0.75, our method yields significantly smaller MSEs compared to the EE method and slightly smaller MSEs than the MLE method. When n=250 and m=750, and λ_n,m = 0.25, our method achieves much smaller MSEs than both the MLE and EE methods. These results underscore the importance of transfer learning methods in the analysis of testing data, when one or more sets of training data are available. Our method effectively incorporates information from the training data to improve estimation and inference for the testing data.








§ A REAL DATA EXAMPLE
 

In this section, we apply our proposed method to study potential risk factors associated with blood pressure for the subcohort of white males whose ages are greater than or equal to 45 and less than 55 (in the range of middle age).


We consider two datasets: the UK Biobank data as the testing data and the data from the Atherosclerosis Risk in Communities (ARIC) study <cit.> as the training data. The UK Biobank is a remarkable resource that has significantly contributed to the field of biomedical research. The database's extensive collection of genetic and health information from a diverse group of half a million UK participants provides researchers with a wealth of data to explore and analyze. Its ongoing expansion and accessibility to approved researchers have facilitated crucial investigations into various diseases and health conditions.
The ARIC study by the National Heart, Lung, and Blood Institute is a comprehensive research initiative with the primary objective of delving into the causes and clinical consequences of atherosclerosis. The ARIC study focuses on studying a diverse group of adults residing in four distinct communities within the United States. By collecting extensive data on these individuals, researchers aim to uncover the underlying factors that contribute to the development and progression of atherosclerosis. This includes investigating genetic, environmental, and lifestyle factors that may influence an individual's susceptibility to atherosclerosis and its associated complications.





Both datasets appear to include several key variables related to health and risk factors. These variables are crucial for studying various aspects of health and disease. Here's a brief summary of the mentioned variables:


  * Systolic Blood Pressure (SBP): This variable measures the pressure in the arteries when the heart beats. It is typically measured in millimeters of mercury (mmHg) and is an important indicator of cardiovascular health.


  * Use of Anti-Hypertensive Medication (HYMD): This binary variable (1 for yes, 0 for no) indicates if the individual is taking medication to manage high blood pressure (hypertension).


  * Glucose Levels (Glucos): This variable measures glucose levels in the blood, often using NMR metabolomics. Glucose is a primary energy source for the body and plays a crucial role in diabetes and metabolic health.


  * Age (Age): Age is a fundamental demographic variable. It is often considered a significant factor in various health outcomes as many conditions are age-dependent.


  * Body Mass Index (BMI): BMI is a measure of body fat based on an individual's weight and height. It is commonly used to assess weight-related health risks.


  * Smoker (Smoker): This binary variable (1 for yes, 0 for no) indicates whether an individual smokes. Smoking is a known risk factor for various health issues, including cardiovascular disease and cancer.


The mention of data cleaning and filtering suggests that the datasets have been processed to remove errors or outliers, ensuring the quality and reliability of the data for research and analysis. Additionally, it's worth noting the sample sizes for each dataset: the UKBiobank dataset has 729 observations, while the ARIC dataset has 786 observations. These observations likely represent unique individuals, and researchers can use this data to explore relationships between these variables and various health outcomes.


As highlighted in the introduction section, model (<ref>) demonstrates remarkable flexibility. It accommodates the presence of selection bias and variations in covariate distributions, which are often attributed to differences in patient inclusion criteria between studies conducted in the UK and the USA. As a consequence, employing model (<ref>) would provide a more reliable approach than assuming that the UK Biobank data and ARIC data share an identical joint distribution.

We use log(SBP) as the continuous response and five other variables as risk factors. It is highly plausible that a prior probability shift issue exists between the UKBiobank and ARIC data due to different study designs and studied populations (USA vs Europe): the averages of SBP in UKBiobank and ARIC data are 135.5 and 115.6, respectively. We apply the three methods from our simulation study to analyze the two datasets: the MLE estimate based on the linear model for the ARIC data (training data); the EE estimate based on the UKBiobank data (testing data); the proposed estimate using both datasets. Table <ref> summarizes the point estimate, the corresponding bootstrap standard error (SE), and the 95% bootstrap percentile confidence interval (BPCI). For comparison, the coefficients for the five risk factors are standardized to have a length of 1.





In Table <ref>, we noticed that the proposed method leads to similar point estimates and detects the same significant risk factors at the 5% level as the MLE method based on the linear model for the ARIC data only, but with much smaller bootstrap SE and shorter BPCIs. The risk factors include HYMD, Glucos, Age, and BMI. In contrast, the EE method based on the UKBiobank data only fails to identify the risk factor HYMD at the 5% level, and furthermore, the lower bound of the 95% CI for the coefficient of the Age variable is close to 0. These results together show the advantage of the proposed method.


In Figure <ref>, the estimated regression function μ_2(t)=μ_2,β(t) in (<ref>),
 derived from the UK Biobank data, is presented. What is evident from this graph is that the relationship between the variables is not linear. This observation suggests that there is a notable difference or shift between the data from the UK Biobank and the ARIC study. Attempting to simply merge or combine these datasets by assuming they are drawn from identical distributions may lead to biased or inaccurate results.

The non-linearity in the estimated regression function suggests that there are likely different patterns or relationships between the variables in the two datasets. It's crucial for researchers to account for these differences and potential biases when conducting analyses or making comparisons between the two datasets. This recognition of dissimilarity can guide the use of more appropriate statistical methods and modeling approaches to address the unique characteristics of each dataset.














§ DISCUSSION
 

In machine learning literature, distribution shifts between the training and testing data are a very common problem. Common distribution shifts include covariate shift and prior probability shift. Dealing with covariate shift is a relatively easy problem since the conditional distribution of Y given X remains the same in both datasets. However, it poses a real challenging problem in the presence of prior probability shift since the conditional density changes from f(y| x) to π(y)f(y| x)/∫π(y)f(y| x)dy, where π(y) is the shift function or selection bias function. This predictive density generally depends on the shift function π(y). Without correcting this selection bias, we end up with a detrimented model.


In this paper, we have introduced a shape-restricted transfer learning approach that effectively merges training and testing data to enhance parameter estimation. This method integrates prior knowledge concerning shape restrictions seamlessly with various practical applications. The challenge of learning monotonic functions has been acknowledged for a long time, particularly when striving for top-tier performance on real-world issues that demand greater flexibility and the assurance of monotonicity across multiple features.

While the monotone single index model has been extensively discussed in statistical and econometric literature over the years, significant breakthroughs in semiparametric maximum likelihood estimation of index coefficients were only recently achieved, particularly by <cit.>. Notably, their work demonstrated that the monotone shape-restricted maximum likelihood of index coefficients is nearly root-n consistent. Moreover, they cleverly constructed a root-n zero-cross estimator using the estimated shape function, adding to the potential of this area of research.



The contributions of this paper open up new avenues for future research in shape-restricted transfer learning and monotonic function estimation, paving the way for improved solutions to various real-world challenges. By building upon these foundations, researchers can further enhance the understanding of complex datasets and unlock even greater potential for practical applications. The findings presented here represent a valuable step forward in the field of statistical and econometric modeling, and we hope they inspire further investigation and innovation in shape-constrained parameter estimation. As more researchers delve into this topic, we anticipate a deeper understanding of the theoretical and practical aspects, ultimately leading to more robust and accurate solutions in various domains.


There are multiple related topics that can be studied subsequently. Firstly, we assumed that the conditional density of Y given X = x between the training and testing data satisfied the relationship f^*(y|x) ∝π(y) f(y|x), and that the effects of X on Y were linear through X^Tβ with β being a common parameter between the training and testing data. It might be of interest to consider other relationships between the conditional densities, such as those explored in <cit.> and <cit.>, and explore the possibility that β might differ across different datasets.
Secondly, our method in this paper is applicable when the dimension p of X is low. We expect that within the framework of this paper, we can extend our method to consider datasets with high-dimensional covariates <cit.>.
Thirdly, we may also extend our method to consider survival data with the response Y being censored.
Lastly, in practice, the covariates of the training data and the testing data may consist of different variables. It could be of interest to consider the transfer learning method on such data.





§ APPENDIX: SKETCH OF THE PROOF FOR PART (3) OF THEOREMS  <REF> AND <REF>



For space limitations, we provide a blueprint of the proof for Part (3) of Theorems <ref> and <ref>. The detailed proof for Parts (1) and (2) of these theorems is presented in the supplementary material.

Besides the notations in Sections <ref> and <ref>, we need the following notations. Denote by F_X_1, Y_1(x, y) and _X_1,Y_1(x, y), respectively, the c.d.f. of (X_1, Y_1) and the empirical c.d.f. of {(X_1,i, Y_1,i}_i=1^n, the observations of the training data. Similarly, we can define F_X_2, Y_2(x, y), _X_2,Y_2(x, y), F_X_1(x), and  F_X_2(x). Let

    ψ(β) = λψ_1(β) + (1-λ) ψ_2(β),

with

    ψ_1(β)    =   ∫μ_1'(x^Tβ)/V(x^T β)x{μ_1(x^Tβ_0) - μ_1 (x^Tβ) } d F_X_1(x), 
    ψ_2(β)    =   ∫x{μ_2,0(x^Tβ_0) - μ_2,β(x^T β) } d F_X_2(x).

They are  the population versions of ψ_n,m(·), ψ_1,n(·), and ψ_2, m(·), respectively defined by (<ref>), (<ref>), and (<ref>). Recalling the definition of φ_β(t) = E(X_2|X_2^Tβ = t), and for any piece-wise constant function μ(·) with finitely many jumps τ_1 <τ_2 < …, we define

    φ̅_β, μ(t) = {[                                        φ_β(τ_i),  if μ_2,β(t)> μ(τ_i),  for all  t ∈ [τ_i, τ_i+1);                                          φ_β(s), if μ_2,β(s) = μ(τ_i),  for some  s∈ [τ_i, τ_i+1);                                      φ_β(τ_i+1),  if μ_2,β(t) < μ(τ_i),  for all  t∈ [τ_i, τ_i+1) ].,

which is a constant function for each i and t ∈ [τ_i, τ_i+1).


We first present two lemmas needed in our subsequent discussion. 

 
Assume Conditions <ref>, <ref>, <ref>, <ref>, and <ref>. We have the following:

    
  (1) ψ(β_0) = 0;
    
  (2) ψ_1'(β_0) = .∂ψ_1(β)/∂β|_β= β_0 exists with rank = p; ψ_2'(β_0) = .∂ψ_2(β)/∂β|_β= β_0 exists with rank = p-1;
    
  (3) We have
    
    ψ_1(β)    =   ψ_1'(β_0)· (β- β_0) + o(β- β_0)
    ψ_2(β)    =   ψ_2'(β_0)·(β - β_0) + o(β - β_0).




  
Assume Conditions <ref>, <ref>, and β_0,1≠ 0. For any λ∈ [0, 1), let 

    B(β_0) =  (  -β_0,2:p^T/β_0,1
     I_p-1),

then ψ_2'(β_0)_-1· B(β_0) has rank p-1.  


Lemma <ref> is Lemma 8, and Lemma <ref> is Lemma 10 in the supplementary material. 


Consider an estimator β̌ of β that satisfies 

    β̌ - β_0 = o_p(1).

We derive the asymptotic properties of 

    ψ_n,m(β̌) = λ_n,mψ_1,n(β̌) + (1-λ_n,m) ψ_2,m(β̌),

where ψ_1,n(·) and ψ_2,m(·) are defined by (<ref>) and (<ref>), respectively. 
Next, we consider the asymptotic properties of ψ_1,n(β̌) and  ψ_2,m(β̌) separately. For ψ_2,m(β̌), we have

    ψ_2,m(β̌)    =   1/m∑_j=1^m X_2,j{Y_2,j-μ_2,β̌(X_2,j^Tβ̌)}
        =    ∫x{y - μ_2,β̌(x^Tβ̌)} d _X_2,Y_2(x, y) 
       =   ∫{x - φ̅_β̌, μ_2, β̌(x^T β̌) }{y - μ_2,β̌(x^Tβ̌)} d _X_2,Y_2(x, y)
       =    J_1 + J_2,

where 

    J_1    =   ∫{x - φ_β̌(x^T β̌) }{y - μ_2,β̌(x^Tβ̌)} d _X_2,Y_2(x, y) ,
    
    J_2    =   ∫{φ_β̌(x^T β̌) -  φ̅_β̌, μ_2, β̌(x^T β̌)}{y - μ_2,β̌(x^Tβ̌)} d _X_2,Y_2(x, y).

In the derivation above, we have used the fact that 

    ∫φ̅_β̌, μ_2, β̌(x^T β̌) {y - μ_2,β̌(x^Tβ̌)} d _X_2,Y_2(x, y) = 0.

This is because φ̅_β̌, μ_2, β̌(x^T β̌) is a piece-wise constant function with the same jumps as μ_2,β̌(t). Recall the definition of μ_2,β̌(t): it is the slope of the greatest convex minorant of the corresponding cusum diagram, based on the values of Y_2,j in the order of X_2,j^Tβ̌ for j=1,…, m. Therefore, on every constant piece [τ_i, τ_i+1), 

    ∫_x^T β̌∈ [τ_i, τ_i+1){y - μ_2,β̌(x^Tβ̌)} d _X_2,Y_2(x, y) = 0.


We summarize the asymptotic properties of J_1 and J_2 in two separate lemmas: Lemmas <ref> and <ref> below.  
Lemma <ref> establishes the asymptotic properties of J_1, while Lemma <ref> considers those of J_2. They are Lemmas 12 and 13 in the supplementary material, and  their proofs are also provided there. 
Combining these two lemmas, we immediately obtain 

    ψ_2,m(β̌) = ψ_2(β̌) + ∫{x - φ_β_0(x^T β_0 ) }{y - μ_2,0(x^Tβ_0)}d_X_2,Y_2(x, y) + o_p(m^-1/2).


 
Assume Conditions <ref>–<ref>, and (<ref>). Referring to J_1 as defined by (<ref>), we have 

    J_1 = ψ_2(β̌) + ∫{x - φ_β_0(x^T β_0 ) }{y - μ_0(x^Tβ_0)}d_X_2,Y_2(x, y) + o_p(m^-1/2).




Assume Conditions <ref>–<ref>, and (<ref>).  Referring to J_2 as defined in (<ref>), we have

    J_2 = o_p(m^-1/2).



With a similar but simpler development as (<ref>) (details can be found in the supplementary material), we obtain 

    ψ_1,n = ψ_1(β̌) + ∫μ_1'(x^Tβ_0)/V(x^Tβ_0)x{ y - μ_1(x^T β_0) } d _X_1, Y_1(x, y) + o_p(n^-1/2).

Combining (<ref>), (<ref>), and (<ref>), we have

    ψ_n,m(β̌)    =   λ_n,mψ_1(β̌) + (1-λ_n,m) ψ_2(β̌) 
           + λ_n,m∫μ_1'(x^Tβ_0)/V(x^Tβ_0)x{ y - μ_1(x^T β_0) } d _X_1, Y_1(x, y) 
          + (1-λ_n,m) ∫{x - φ_β_0(x^T β_0 ) }{y - μ_2,0(x^Tβ_0)}d_X_2,Y_2(x, y) 
          + λ_n,m·  o_p( n^-1/2) + (1-λ_n,m) ·  o_p(m^-1/2) 
       =   λ_n,mψ_1(β̌) + (1-λ_n,m) ψ_2(β̌) 
           + λ_n,m∫μ_1'(x^Tβ_0)/V(x^Tβ_0)x{ y - μ_1(x^T β_0) } d _X_1, Y_1(x, y) 
          + (1-λ_n,m) ∫{x - φ_β_0(x^T β_0 ) }{y - μ_2,0(x^Tβ_0)}d_X_2,Y_2(x, y) 
          +  o_p((n+m)^-1/2).


With the same arguments as those in Section 4 of the supplementary material, we have consistent estimators of β: β^L and β^U, and 0 ≤α_n,m≤ 1 (in the proof for Part (1) of Theorems <ref> and <ref>). Furthermore, for λ > 0, based on (<ref>) and Lemma <ref>, we have

    0   =   α_n,mψ_n,m(β^L) + (I_p-α_n,m) ψ_n,m(β^U) 
        =    α_n,m{λ_n,mψ_1(β^L) + (1-λ_n,m) ψ_2(β^L)}
           + (I_p-α_n,m) {λ_n,mψ_1(β^U) + (1-λ_n,m) ψ_2(β^U)}
           + λ_n,m∫μ_1'(x^Tβ_0)/V(x^Tβ_0)x{ y - μ_1(x^T β_0) } d _X_1, Y_1(x, y) 
          + (1-λ_n,m) ∫{x - φ_β_0(x^T β_0 ) }{y - μ_2,0(x^Tβ_0)}d_X_2,Y_2(x, y) 
          +  o_p((n+m)^-1/2) 
       =    A(β_0) (β- β_0)  + λ_n,m∫μ_1'(x^Tβ_0)/V(x^Tβ_0)x{ y - μ_1(x^T β_0) } d _X_1, Y_1(x, y) 
          + (1-λ_n,m) ∫{x - φ_β_0(x^T β_0 ) }{y - μ_2,0(x^Tβ_0)}d_X_2,Y_2(x, y) 
          +  o_p((n+m)^-1/2) + o_p(β- β_0) 
       =    A(β_0) (β- β_0) + λ_n,m^1/2 (n+m)^-1/2 G_1,n + (1-λ_n,m)^1/2 (n+m)^-1/2 G_2,m
           +  o_p((n+m)^-1/2) + o_p(β- β_0),

where A(β_0) = λψ_1'(β_0) + (1-λ) ψ_2'(β_0),

    G_1,n   =   √(n)∫μ_1'(x^Tβ_0)/V(x^Tβ_0)x{ y - μ_1(x^T β_0) } d _X_1, Y_1(x, y),
    
    G_2,m   =   √(m)∫{x - φ_β_0(x^T β_0 ) }{y - μ_2,0(x^Tβ_0)}d_X_2,Y_2(x, y).

Based on Conditions <ref>–<ref> and the Central Limit Theorem, we have

    G_1,n→    N(0, Σ_1 ), 
    and    G_2,m→    N(0, Σ_2),

in distribution, where 

    Σ_1    =   var[ μ_1'(X_1^Tβ_0)/V(X_1^Tβ_0) X_1 { Y_1 - μ_1(X_1^T β_0) }]
    Σ_2    =   var[ {X_2 - φ_β_0(X_2^T β_0 ) }{Y_2 - μ_2,0(X_2^Tβ_0)}].

We, therefore, conclude that the c.d.f. of (G_1,n^T, G_2,m^T)^T is given by:

    P( (G_1,n^T, G_2,m^T)^T ≤ (g_1^T, g_2^T)^T ) 
        =     P(G_1,n≤ g_1) P( G_2,m≤ g_2 ) 
       →   Φ_Σ_1(g_1) Φ_Σ_2(g_2),

for any g_1, g_2 ∈^p, where Φ_Σ(·) denotes the c.d.f. for the N(0, Σ) distribution. This implies

    ( G_1,n
     G_2,m) → N(0, (Σ_1     0 
     0    Σ_2 ) ),

in distribution. Combining (<ref>) with (<ref>), applying Delta method, and based on the fact that λ_n,m→λ, we conclude 

    √(n+m)( β- β_0 ) → N(0, A_0(β_0)^-1{λΣ_1 + (1-λ) Σ_2 } A_0^-1(β_0) ),

in distribution, where 

    A(β_0)     =    -λψ_1'(β_0) - (1-λ) ψ_2'(β_0) 
       =   λ E[ {μ_1'(X_1^Tβ_0)}^2/V(X_1^T β_0) X_1 X_1^T ] 
           + (1-λ)E[ {X_2- φ_β_0(X_2^T β_0)}{X_2 - φ_β_0(X_2^T β_0)}^T μ_2,0'(X_2^Tβ_0 )   ].


For λ = 0, since β is a zero crossing of ψ_n,m(β)_2:p,  based on (<ref>) and Lemma <ref>, we have 

    0   =   α_n,mψ_n,m(β^L)_2:p + (I_p-1-α_n,m) ψ_n,m(β^U)_2:p
        =    α_n,m{λ_n,mψ_1(β^L) + (1-λ_n,m) ψ_2(β^L)}_2:p
           + (I_p-α_n,m) {λ_n,mψ_1(β^U) + (1-λ_n,m) ψ_2(β^U)}_2:p
           + λ_n,m∫μ_1'(x^Tβ_0)/V(x^Tβ_0)x_2:p{ y - μ_1(x^T β_0) } d _X_1, Y_1(x, y) 
          + (1-λ_n,m) ∫{x_2:p - φ_β_0(x^T β_0 )_2:p}{y - μ_2,0(x^Tβ_0)}d_X_2,Y_2(x, y) 
          +  o_p((n+m)^-1/2) 
       =   λ_n,mψ_1(β)_2:p + (1-λ_n,m) ψ_2(β)_2:p
           + λ_n,m^1/2 (n+m)^-1/2G_1,n + (1-λ_n,m)^1/2 (n+m)^-1/2G_2,m
          +  o_p((n+m)^-1/2) 
       =    - A(β_0)_2:p B(β_0) (β_2:p - β_0, 2:p)  
           + λ_n,m^1/2 (n+m)^-1/2G_1,n + (1-λ_n,m)^1/2 (n+m)^-1/2G_2,m
          +  o_p((n+m)^-1/2) + o_p(β_2:p - β_0, 2:p)
       =   ψ_2'(β_0)_2:p B(β_0) (β_2:p - β_0, 2:p)  
           + λ_n,m^1/2 (n+m)^-1/2G_1,n + (1-λ_n,m)^1/2 (n+m)^-1/2G_2,m
          +  o_p((n+m)^-1/2) + o_p(β_2:p - β_0, 2:p),

where 

    G_1,n   =   √(n)∫μ_1'(x^Tβ_0)/V(x^Tβ_0)x_2:p{ y - μ_1(x^T β_0) } d _X_1, Y_1(x, y), 
    G_2,m   =   √(m)∫{x_2:p - φ_β_0(x^T β_0 )_2:p}{y - μ_2,0(x^Tβ_0)}d_X_2,Y_2(x, y),
    
     B(β_0)    =   (  -β_0,2:p^T/β_0,1
     I_p-1).

Based on Lemma <ref>, ψ_2'(β_0)_2:p B(β_0) is of full rank. Based on Conditions <ref>–<ref>, we have G_1,n = O_p(1), and

    λ_n,m^1/2 (n+m)^-1/2G_1,n   =    o_p((n+m)^-1/2) 
    G_2,m   →     N(0, Σ_2),

with

    Σ_2    =   var[ {(X_2)_2:p - φ_β_0(X_2^T β_0 )_2:p}{Y_2 - μ_2,0(X_2^Tβ_0)}] = (Σ_2 )_-1,-1.


Combining (<ref>) with (<ref>) leads to 

    √(n+m)(β_2:p - β_0, 2:p)→{A(β_0) }^-1 N(0, Σ_2),

in distribution, where with straightforward evaluations, we have,

    A(β_0)     =    -ψ_2'(β_0)_2:p B(β_0) 
       =    E[ {X_2 - φ_β_0(X_2^T β_0)}_2:p{X_2 - φ_β_0(X_2^T β_0)}_2:p^T μ_2, 0'(X_2^Tβ_0 )   ] 
          ×(β_0,2:pβ_0,2:p^T + I_p-1).

We complete the proof for Part (3) of Theorems <ref>
and <ref>. 


apalike




 Paper Title: Conditional Independence in Stationary Diffusions.



















§ ABSTRACT 
Stationary distributions of multivariate diffusion processes have
recently been proposed as probabilistic models of causal systems in statistics and machine learning. Motivated by these developments, we study stationary multivariate diffusion processes with a sparsely structured drift. Our main result gives a characterization of the conditional independence relations that hold in a stationary distribution. The  result draws on a graphical representation of the drift structure and pertains to conditional independence relations that hold generally as a consequence of the drift's sparsity pattern.



Constraints on large-scale polarization in northern hemisphere
    Chang Feng
    August 5, 2024
==============================================================













§ INTRODUCTION
 




Consider an ℝ^-valued stationary stochastic process {𝕏(t)}_t≥ 0 that arises as the solution 
of a stochastic differential equation (SDE)

    d𝕏(t)= f(𝕏(t)) dt + D  d𝕎(t),

 understood in the Itô sense <cit.>.
Here,  f: ℝ^→ℝ^ is a Lipschitz-continuous map that specifies the drift, D∈ℝ^× is a diagonal and invertible matrix, and 𝕎 is a standard -dimensional Brownian motion.  Assuming its unique existence, the stationary (marginal) distribution of the process 𝕏 depends on D only through the diffusion matrix C =  DD^T, and we denote the distribution by P_f,C.  Our interest lies in examining the conditional independence (CI) relations that hold in P_f,C when the drift is sparse, i.e., when each component function f_i only depends on some of its input coordinates.

Let X be an ℝ^-valued random vector  distributed according to P_f,C.  Let I,J,K⊆[]:={1,…,} be disjoint index sets, and denote subvectors of X as X_I=(X_i)_i∈ I.  Then we ask which sparsity restrictions on f imply that X_I and X_J are conditionally independent given X_K; in symbols, X_I X_J| X_K or briefer I J| K.  The main results of this paper answer this question via a characterization based on a directed graph that encodes the sparsity assumptions about the map f.










Before proceeding, we firm up the framework.

If f is Lipschitz-continuous, the SDE in (<ref>), initialized at a random vector 𝕏(0) that is independent of the driving Brownian motion, has a unique but not necessarily stationary solution.



Existence and uniqueness of a stationary solution may be guaranteed in the framework provided by <cit.>.  To this end, we assume that f has component functions f_i that are locally m-integrable for m>, and that f satisfies a Lyapunov condition. Referring to the entries of the diagonal diffusion matrix C =  (c_ij), define the  adjoint Fokker–Planck operator 

    ℒ = ∑_i=1^1/2c_ii∂_i^2 + f_i ∂_i.

 The Lyapunov condition then requires existence of a non-negative function U ∈ C^2(ℝ^), the Lyapunov function, fulfilling
 
    lim_x→∞ U(x) = ∞,  and lim sup_x→∞ℒU(x) < 0.








 
In the sequel, we use graphs to represent the sparsity structure of a drift map f.
Specifically, we consider directed graphs = (V, E) whose vertex set V= [] indexes the coordinate processes of 𝕏.
For all graphs considered in the paper, we  tacitly assume that the edge set includes all self-loops i→ i.

Let _(i) = {j ∈[]: (j,i) ∈ E} denote the set of parents of node i∈ V in graph .  By our convention on self-loops, i∈_(i).


We call a drift map f:ℝ^→ℝ^ -compatible  if it is Lipschitz-continuous, satisfies the above local integrability and Lyapunov conditions, and has its sparsity pattern given by , that is, each coordinate function f_i:ℝ^→ℝ only depends the values x__(i) of input coordinates indexed by _(i), i∈[].




We now introduce our main theorem.  If there is a directed path from node i to node j in the directed graph , then i is an ancestor of j.  Write _(i) for the set of all ancestors of i.  Note that i∈_(i).  
Given , we construct an auxiliary graph =(V,Ê) that we call the trek graph.  It is obtained by including a bidirected edge i↔ j in Ê if _(i)∩_(j)≠∅. 


    
    Let =(V,E) be a directed graph on V=[], and let  be its trek graph.  Let I,J,K⊆[] be three disjoint index sets.  Then the following two statements are equivalent: 
    
        
  (i) The set I is separated from J by V ∖ (I ∪ J ∪ K) in the trek graph .
        
  (ii) The conditional independence X_I  X_J | X_K holds in any random vector X that is distributed according to a stationary distribution P_f,C given by a -compatible drift map f:ℝ^→ℝ^ and a positive definite diagonal matrix C∈ℝ^×. 
    


    




   Consider the graph  depicted in <Ref>, and let X be a random vector following a stationary distribution  given by a -compatible drift map and a positive definite diagonal diffusion matrix. The graph  has the ancestor sets,
   
   _(1)   ={1},      _(2)   ={1,2},       _(3)   ={1,2,3,4},      _(4)   ={4}.
   
   As only the pairs of nodes (1,4) and (2,4) do not have a common ancestor, the trek graph  is missing precisely the edges 1 ↔ 4 and 2 ↔ 4; see <Ref>.  For subsets of nodes I, J, and S, write I⊥_ J| S if I and J are separated by S in  (i.e., every path from a node i∈ I to a node j∈ J intersects S).  Then <Ref> implies the following marginal and conditional independencies:
   
       {1,2}   ⊥_{4}|{3}              X_{1,2}    X_4,

       {1}   ⊥_{4}|{2,3}              X_1    X_4,

       {2}   ⊥_{4}|{1,3}              X_2    X_4,

       {1}   ⊥_{4}|{3}              X_1    X_4|X_2,

       {2}   ⊥_{4}|{3}              X_2    X_4|X_1.
   
   Here, the first listed separation implies the other separations graphically, and the first listed marginal independence implies the other independencies probabilistically by the semi-graphoid axioms of conditional independence <cit.>. 

    <Ref> further states that no marginal or conditional independencies  other than the ones listed above hold for all considered stationary distributions. For instance, <Ref> implies existence of a -compatible drift map f and a positive definite diagonal diffusion matrix C such that 
    
    X_1 X_4 | X_3

    for X ∼ P_f,C. We demonstrate how to construct such distributions in <Ref>.


The proof of <Ref> will be developed in the subsequent sections.  The fact that separation in the trek graph implies conditional independence will be derived with the help of the Fokker–Planck equation that characterizes the density of P_f,C (<Ref>). In constructing counterexamples that exhibit conditional dependence when separation fails, we strive hard to structure them as simple as possible.  Specifically, our counterexamples take f to be linear, which gives an Ornstein–Uhlenbeck process for which we let the diffusion matrix be a multiple of the identity (<Ref>). 


Stationary diffusion processes have emerged as a new framework for statistical analysis of causal systems.  Indeed, in many applications it is difficult if not impossible to observe a system more than once; e.g., in biology, measurement may destroy the observed collection of cells. 
The available data then constitute a sample of independent observations, for which the new paradigm retains an implicit temporal perspective by considering one-time snapshots of independent stationary dynamic processes. This approach was pioneered using stationary Ornstein–Uhlenbeck processes, resulting in Gaussian models termed Lyapunov models <cit.>.  Research progresses on issues like identifiability of drift parameters, statistical estimation, and new applications <cit.>.  Moreover, extensions to non-linear diffusions were proposed in <cit.>.  <Ref> advances this line of research via a characterization of conditional independence  that can be used to disprove parameter identifiability <cit.>. It also reveals that even for general drift maps, the trek graph  may be estimated by  learning a marginal independence graph <cit.>.



Working with stationary distributions of sparse multivariate diffusion processes has particular appeal when modeling causal systems with feedback loops, which present mathematical and interpretational challenges for the classical structural equation models (SEMs) <cit.>. For the latter, the well-known d-separation criterion yields a graphical characterization of conditional independence relations <cit.>. In our <Ref>, a marginal independence X_I X_J (with conditioning set K=∅) arises from the absence of common ancestors of nodes in I and nodes in J, which is precisely the same phenomenon as in d-separation. However, as will be evident from our later proof, <Ref> shows that for stationary diffusions the marginal independencies are the only source of independence relations, which radically differs from the case of d-separation.
















The remainder of the paper is organized as follows.

<Ref> treats the characterization of marginal independence and describes how to infer the conditional independence relations implied by these marginal independencies by means of a connected set Markov property for the trek graph.

In <Ref>, we set up notation, formally introduce the Lyapunov model and review some facts on Gaussian distributions. 

In <Ref>, we prove the main result by showing that all conditional independence relations that hold in the Lyapunov model originate from marginal independencies.  To this end, we reduce the problem to graphs obtained from 1-clique sums of treks. We then use a combination of algebraic and analytic methods to establish the result in this simpler case.





§ MARGINAL INDEPENDENCE AND IMPLIED CONDITIONAL INDEPENDENCIES



This section proves one direction in <Ref>, namely, that V ∖ (I ∪ J ∪ K) separating I and J in the trek graph  implies the conditional independence X_I  X_J | X_K.  We start with the case K=∅, i.e., marginal independence statements. Then, we show that these statements imply all claimed conditional independence statements. Before giving the proofs, we introduce graphical concepts, as used in this as well as later sections of the paper.



 §.§ Graphs
 

For a directed graph = (V, E) with edge set E ⊆ V × V, we write i → j if (i,j) ∈ E.  We denote the set of nodes or edges of a specific graph  by V() or E(), respectively.  The set _(i) = {j ∈ V: j → i} collects the parents of a node i∈ V, and _(i) = {j ∈ V: i → j} is its set of children. 
A directed path is a sequence of 

nodes (i_1, …, i_m) such that i_l → i_l+1∈ E for 1≤ l<m, and m-1 is the length of this path. A path of length 0 consists of a single node.
As defined in the introduction, _(i) is the set of ancestors of i, i.e., the nodes j from which there is a directed path to i.
This definition is conjunctively applied to sets, so _(I) = ⋃_i ∈ I_(i). The set of descendants of a node i is denoted _(i) and consists of all nodes j such that there is a directed path from i to j. Typically, it is clear which graph  is concerned, and we  drop the subscript, writing (i)≡_(i) and, similarly,  (i) and (i). The respective sets where the node itself is excluded are denoted in lowercase, e.g., (i)=(i)∖{i} and (i)=(i)∖{i}.

Combining two directed paths yields a trek: An (ℓ,r)-trek between nodes i_ℓ and j_r is a sequence of 

nodes (i_ℓ, …, i_1, t, j_1, …, j_r) such that both (t, i_1, …, i_ℓ) and (t, j_1, …, j_r) are directed paths.  Node t is the top node of the trek.  If there is a trek between two nodes i and j, then the top node is a common element of (i) and (j).  Thus, the trek graph  from <Ref> has two nodes adjacent precisely if there is a trek between them.

A subset of nodes I ⊆ V is ancestral if (I) = I. A trivial example of an ancestral set is a source node, meaning a node without parents.  
If a directed path starts and ends at the same node, then it forms a directed cycle. If the graph  does not contain any directed cycles, then it is termed a directed acyclic graph (DAG).  The vertex set of a DAG may be ordered topologically as (i_1, …, i_n) such that for l ≠ k, i_l∈(i_k) implies that l<k.










 §.§ Marginal Independence


The first result treats the implication (i)⇒(ii) in <Ref> when the conditioning set K is empty, in which case the independence is marginal.

 
Let 𝒢=(V,E) be a directed graph on V=[]. If f is 𝒢-compatible, then X ∼ P_f, C satisfies X_I  X_J for all sets of nodes I, J ⊆ V without trek between them.

This theorem generalizes an observation concerning Ornstein–Uhlenbeck processes made in <cit.>, and we prove it using the Fokker–Planck equation.  To this end, we draw on the next lemma, which follows from combining Theorem A and Proposition 2.1 in <cit.>.  Here, C_0^∞(ℝ^) is the set of smooth functions with compact support.

 If the Lyapunov condition is fulfilled, then (<ref>) has a unique stationary solution admitting a density p. This density is the unique density fulfilling the Fokker–Planck equation
  
    ∫_ℝ^ℒϕ(x)p(x) dx = 0  for all ϕ∈ C_0^∞(ℝ^) .
 
 



 
By <Ref>, the stationary distribution P_f, C has a density p.
We begin by showing that for any ancestral set A⊆ V, the marginal density p_A of X_A satisfies the Fokker–Planck equation for the SDE restricted to A. Let A^ = V ∖ A be the set of all remaining nodes. Fix a function ϕ∈ C_0^∞(ℝ^) with ϕ(x) only depending on x_A. Then the Fokker–Planck equation for the overall density p yields that

    0 =  ∫_ℝ^ℒϕ(x)p(x) dx 
     =∫_ℝ^( ∑_i=1^1/2c_ii∂_i^2 ϕ(x) + f_i(x) ∂_i ϕ(x) ) p(x) dx 
    
     = ∫_ℝ^( ∑_i∈ A1/2c_ii∂_i^2 ϕ(x) + f_i(x) ∂_i ϕ(x) ) p(x) dx
 
because ∂_i ϕ(x) ≡ 0 for i ∉ A as ϕ only depends on x_A. Now, the ancestrality of A implies that f_i for i∈ A only depends on x_A.  Rewriting the above integral, we obtain, by Fubini,

    0 
     
     
      =    ∫_ℝ^A∫_ℝ^A^( ∑_i∈ A1/2c_ii∂_i^2 ϕ(x) + f_i(x) ∂_i ϕ(x) ) p(x) dx_A^ dx_A 
    
      =    ∫_ℝ^A( ∑_i∈ A1/2c_ii∂_i^2 ϕ(x) + f_i(x) ∂_i ϕ(x) ) ∫_ℝ^A^p(x) dx_A^ dx_A 
    
      =    ∫_ℝ^A( ∑_i∈ A1/2c_ii∂_i^2 ϕ(x_A) + f_i(x_A) ∂_i ϕ(x_A) ) p_A(x_A)  dx_A,
 
  which is the Fokker–Planck equation for a restricted SDE of the form
  
    d𝕏_A(t) = f_A(𝕏_A(t))  dt + D_A, A d𝕎(t)

    with D_A,AD_A,A^T=C_A,A.
Applying this result to A=(I) ∪(J) gives a Fokker–Planck equation for the marginal density of  X_(I) ∪(J). To show the independence statement, we show that the product of marginal densities p_(I) p_(J) fulfills the same Fokker–Planck equation. Let Ĩ = (I), J̃ = (J), and ϕ∈ C_0^∞(ℝ^A). Since there is no trek between I and J, the sets Ĩ and J̃ are disjoint and partition A=Ĩ∪J̃. Therefore,

    ∫_ℝ^A( ∑_i∈ A1/2c_ii∂_i^2 ϕ(x_A) + f_i(x_A) ∂_i ϕ(x_A) )(p_Ĩ p_J̃)(x_A) dx_A 
     =   ∫_ℝ^A( ∑_i∈Ĩ1/2c_ii∂_i^2 ϕ(x_A) + f_i(x_A) ∂_i ϕ(x_A) )(p_Ĩ p_J̃)(x_A) dx_A 
       +∫_ℝ^A( ∑_i∈J̃1/2c_ii∂_i^2 ϕ(x_A) + f_i(x_A) ∂_i ϕ(x_A) )(p_Ĩ p_J̃)(x_A) dx_A.

We rewrite the first integral as

    ∫_ℝ^A( ∑_i∈Ĩ1/2c_ii∂_i^2 ϕ(x_A) + f_i(x_A) ∂_i ϕ(x_A) )(p_Ĩ p_J̃)(x_A) dx_A 
     =   ∫_ℝ^J̃∫_ℝ^Ĩ( ∑_i∈Ĩ1/2c_ii∂_i^2 ϕ(x_A) + f_i(x_A) ∂_i ϕ(x_A) )(p_Ĩ p_J̃)(x_A) dx_Ĩ dx_J̃
     
     =   ∫_ℝ^J̃∫_ℝ^Ĩ( ∑_i∈Ĩ1/2c_ii∂_i^2 ϕ(x_A) + f_i(x_A) ∂_i ϕ(x_A) )p_Ĩ (x_Ĩ)  dx_Ĩ  p_J̃(x_J̃)  dx_J̃.
 
  The Fokker–Planck equation for Ĩ, with test function x_Ĩ↦ϕ(x_Ĩ,x_J̃) for fixed x_J̃, yields that the inner integral vanishes. Hence, the overall integral vanishes as well. Reversing the roles of I and J shows that also the second summand in (<ref>) is zero.  In conclusion, p_(I) p_(J) fulfills the Fokker–Planck equation for p_(I)∪(J), i.e., p_(I) p_(J) is a density of X_(I)∪(J).  Consequently, X_(I) X_(J) and the decomposition property of conditional independence implies our claim that X_I X_J because I⊆(I) and J⊆(J).




 §.§ Implied Conditional Independencies












   




Let =(V,Ê) be the trek graph associated to the given directed graph .  By definition, (i,j)∈Ê if and only if there is a trek between nodes i and j.  Observe that (i,j) ∈Ê if and only if (j,i) ∈Ê, and, reflecting the structure of treks, we draw the edge bidirected as i↔ j.  <Ref> clarifies that the trek graph encodes marginal independencies, in particular,

    i↔ j ∉Ê X_i X_j

in random vectors X∼ P_f,C with f a -compatible drift map.
A more refined interpretation of the trek graph allows us to derive all conditional independencies that are implied by the marginal statements from <Ref>.   Recall the notation I⊥_ J| S  from <Ref>, which indicates that the subsets of nodes I and J are separated by subset S in .







Let =(V,E) be a directed graph on V=[], and consider a random vector X∼ P_f,C, where f:ℝ^→ℝ^ is a -compatible drift map and C∈ℝ^× is positive definite and diagonal.  Let I,J,K⊆[] be three disjoint index sets, and let  be the trek graph of .   Then 

    I⊥_ J| V ∖ (I ∪ J ∪ K)     
    X_I  X_J | X_K.






In the literature on probabilistic graphical models, a random vector X that satisfies (<ref>) is said to satisfy the pairwise Markov property relative to the bidirected graph  <cit.>.  However, <Ref> shows that a richer set of marginal independencies hold.  Call a subset A⊆ V connected in  if any two vertices i, j ∈ A are connected by a path in  all of whose vertices are in A (i.e., A induces a connected subgraph of ).  Let 
Sp_(A) be the set of all nodes that are in A or are adjacent to an element of A via an edge of .  Then <Ref> implies that X satisfies the connected set Markov property relative to , which requires that 

    X_A  X_V ∖Sp_(A)  ∀  A⊂ V,  non-empty and connected in .

By general properties of conditional independence, the connected set Markov property is equivalent to the global Markov property for bidirected graphs <cit.>.  The global Markov property states that 

    I⊥_ J| V ∖ (I ∪ J ∪ K)     
    X_I  X_J | X_K,

as was the claim of our theorem.



We illustrate the theorem in another example.




Let  be the bidirected graph from <Ref>.  The connected set Markov property requires

    1{3,4},  {1,2} 4.

As 1 is separated from 4 by 3 in , the global Markov property additionally states 1  4 | 2. The other conditional (and not marginal) independencies required by the global Markov property are

    1 3 | 4,   1 4 | 3,   2 4 | 1.




In the next sections, we show that <Ref> already captures all conditional independencies that hold for general -compatible drift. To this end, we consider Ornstein–Uhlenbeck processes and the graphical Lyapunov models obtained from their stationary distributions.













§ ORNSTEIN–UHLENBECK PROCESSES AND LYAPUNOV MODELS



This section briefly reviews Ornstein–Uhlenbeck processes and the associated graphical continuous Lyapunov models. We also recall basic facts regarding Gaussian conditional independence that will be used in <Ref>.



 §.§ Ornstein–Uhlenbeck Processes



The Ornstein–Uhlenbeck process is the only non-trivial time-homogeneous Markov process that is both Gaussian and stationary <cit.>.  In the -dimensional case, it is defined by the SDE

    d𝕏(t) = M (𝕏(t)-μ)  dt + D  d𝕎(t)

with an invertible drift matrix M ∈^× encoding the interactions among the coordinates of 𝕏(t). 
The affine linear drift term introduces a restoring force towards the mean vector μ.

Again we only consider the case with D and the diffusion matrix C=D D^T diagonal.


Evidently, (<ref>) is a special case of
(<ref>) with f being an affine
function given by M and μ.  In this case, the Lyapunov condition
on f is satisfied if M is a stable matrix, that is, the
real parts of the eigenvalues of M are negative.  For M stable,
𝕏(t) admits a unique stationary distribution that is Gaussian with mean μ <cit.>.  The positive definite covariance matrix of the distribution is the unique solution Σ≡Σ(M,C) of the continuous Lyapunov equation

    MΣ + Σ[-0]M = -C.






We note that the solution Σ is invariant under joint rescaling of M and C. 

Given our interest in conditional independence structure, we restrict
ourselves, without loss of generality, to stationary distributions of
Ornstein–Uhlenbeck processes with mean μ = 0. For a directed
graph , the class of Gaussian distributions of interest 
naturally corresponds to the set of covariance matrices Σ
that can be achieved using stable matrices M for which the linear
map x ↦ Mx is -compatible.  In other words, we consider
stable matrices M whose support (i.e., non-zero entries) is
contained in the edge set E(). 






    Let = (V, E) be a directed graph on V = []. Let ^E be the set of matrices M = (m_ij) ∈^× such that m_ji = 0 if i → j ∉ E and denote by _ the cone of × positive definite matrices. Then for any fixed C ∈_, the (graphical continuous) Lyapunov model of  is the family of all multivariate Gaussian distributions (0,Σ) with covariance matrix Σ in the set
    
    _,C = {Σ∈_ :  M Σ + Σ M^T + C = 0  for a stable matrix  M ∈^E}.


In the sequel, we identify the model with its set  of covariance matrices and refer to _,C as the Lyapunov model. We denote the entries of a covariance matrix Σ by σ_ij. 


We denote the -dimensional identity matrix by _ or equivalently by _V for a set V with |V| =. We will typically fix C = 2 _ and write _ = _, 2 _. This assumption guarantees that _∈_, C which is a desirable property for any Gaussian model. While the drift matrix M is typically thought of as the weighted adjacency matrix of , we will usually suppress the self-loops in subsequent drawings of graphs for simplicity. 







 §.§ Conditional Independence in Multivariate Gaussians






Consider a symmetric matrix Σ with rows and columns indexed by a finite set V. For I,J⊆ V, let Σ_I,J denote the submatrix with rows indexed by I and columns by J. Submatrices of the form Σ_K Σ_K,K are principal. 

Especially in subscripts, we will often abbreviate a union I ∪ J of index sets to IJ. A single index i is used interchangeably with its singleton set {i} in these conventions.
So, iK{i}∪ K and K ∖ i K ∖{i}.


The starting point of our subsequent study of conditional independence is the following collection of results. Proofs can be found, for instance, in 4.1 of <cit.>.














 
Let Σ∈_n.
The conditional independence (CI) statement I  J | K holds in the Gaussian random vector X ∼(μ, Σ) if and only if submatrix
Σ_IK, JK has rank |K|. 


Since Gaussian distributions satisfy the compositional graphoid axioms, the CI statement I  J | K holds if and only if i  j | K holds for all i ∈ I and j ∈ J <cit.>. Thus, to characterize the set of CI statements that a Gaussian distribution satisfies, it suffices to characterize all statements of the form i  j | K. This is summarized with the following corollary, which combines this fact with the previous theorem. 




For a Gaussian random vector X∼(μ, Σ) with Σ∈_n, the following statements are equivalent:


  * I  J | K,

  * i  j | K for all i ∈ I and j ∈ J, and

  * (Σ_iK, jK) ≡ |Σ_iK, jK| = 0 for all i ∈ I and j ∈ J.



This characterization of conditional independence for multivariate Gaussians will be used extensively in <Ref>.






























§ CONDITIONAL INDEPENDENCE IN LYAPUNOV MODELS



Our focus is now on the 

Lyapunov model of a directed graph = (V,E), which is given by the set of covariance matrices 

    _ = {Σ∈_ :  M Σ + Σ M^T + 2 _ = 0   for a stable matrix  M ∈^E}.


<ref> applies to _ 
and shows that the absence of a trek in  leads to a marginal independence, and <ref> yields the implied 

additional conditional independence (CI) statements for _.
The aim of this section is to prove that the resulting collection of CI statements is complete, by showing that for any other CI statement, there is a distribution Σ∈_ which violates it.
More precisely, our main result is as follows.






Let  be a directed graph with trek graph . Then the Lyapunov model _ is Markov-perfect to the marginal independence graph . In other words, i  j | K holds for all Σ∈_ if and only if  V ∖ ({i ,j }∪ K) separates i and j in . 


The proof of this theorem will be completed at the end of this section. As noted above, the direction to be proven is the fact that a CI statement holds for all Σ∈_ only if we have graphical separation.




 §.§ Outline of the Proof



We will prove the contrapositive, 
i.e., we will show that if V ∖ ({i ,j }∪ K) does not separate i and j in , then there exists a matrix Σ∈_ such that i j | K.  To do this, we first clarify that we may reduce the problem to studies of suitable subgraphs, and we introduce a trek rule which gives a polynomial parameterization of a submodel of _ in terms of the entries of M (<ref>). We then show how to apply this restricted trek rule to the case where  is a single trek in order to produce covariance matrices Σ in _ such that the endpoints of the trek are conditionally dependent given all other nodes on the trek 
Via a trick of creating a limiting scenario of perfect correlation, we  extend this result to more general conditioning sets comprised of arbitrary subsets of the other nodes on the trek (<ref>).

Finally, to complete the proof, we extend the previous results from a single trek to zig-zag graphs, which correspond to paths that are connecting for the global Markov property described in the proof of <ref> (<ref>).  We note that these zig-zag graphs also appear in the description of the d-separation criterion for structural equation models; cf. <cit.>. 



 §.§ Subgraphs and a Restricted Trek Rule



The following lemma clarifies how to pass to subgraphs for the construction of counterexamples to CI statements.


 
Let = (V, E) be a directed graph, and let ' = (V', E') be a subgraph of . 
If i j | K' on _', then i j | K on _ for all K ⊆ V with K ∩ V' = K'.



The hypothesis that i j | K' means that there exists Σ' ∈_' for which |Σ'_iK',jK'| ≠ 0. Let M' be the corresponding parameter matrix, i.e., M'Σ' + Σ' M'^T + 2 _V' = 0. Denote V” = V ∖ V'. Then the matrix equation with blocks indexed by V' and V”

    [   M'    0;    0 -_V” ]_ M[  Σ'   0;   0 _V” ]_Σ + 
      [  Σ'   0;   0 _V” ]_= Σ[ M'^T    0;    0 -_V” ]_= M^T +
      [ 2 _V'     0;     0 2 _V” ]_= 2 _V = 0

shows that Σ∈_. Since K ∩ V' = K', it follows that |Σ_iK,jK| = |Σ'_iK',jK'| ≠ 0.


The previous lemma will be used throughout this section to reduce the problem of determining whether a conditional dependence holds to a minimal subgraph which exhibits it. The next proposition is then our major tool to study the resulting minimal subgraphs.




Let = (V, E) be a DAG, and consider the slice of the model _ obtained by restricting the drift matrix M=(m_ij) to have diagonal entries m_ii = -12ζ for all i ∈ V. Then the following trek rule holds for the entries of the covariance Σ=(σ_ij) associated to M:

    σ_ij = ∑_T : (ℓ, r)-trek
    from i to j 2 ζ^ℓ+r+1ℓ+rℓ m_T,

where the trek monomial m_T associated to a trek T is given by m_T = ∏_e ∈ Tm_e, i.e., the product over all the edges e in that trek.




The Lyapunov equation MΣ + Σ M^T = -C is linear in Σ, and 












the ij-th element of the matrix equation reads

    (m_ii + m_jj) σ_ij + ∑_l ∈(j) m_jlσ_li + ∑_k ∈(i) m_ikσ_kj = -C_ij.

Note that the coefficient m_ii + m_jj of σ_ij is always -1ζ. Solving for σ_ij, we obtain

    σ_ij = ζ·(C_ij + ∑_l ∈(j) m_jlσ_li + ∑_k ∈(i) m_ikσ_kj).

These relations motivate a proof of (<ref>) by induction. 

We fix a topological ordering of the DAG  and extend it to a lexicographic ordering on the entries of Σ via σ_ij < σ_kl if and only if j < l or j = l and i < k. We perform induction with respect to this ordering, noting that the right-hand side of (<ref>) contains only Σ-entries which are strictly smaller than σ_ij on the left-hand side.

The smallest Σ-entries are σ_ii where i is a source node. In this case σ_ii = ζ C_ii = 2ζ follows immediately from the Lyapunov equation for our choice of C = 2_V. This verifies the claim as there is only the empty trek on i. <Ref> or equation (<ref>) can be used to see that if i and k are distinct source nodes (having in particular no treks between them), then σ_ki = 0. Now let j be a node all of whose parents are source nodes; the edges into such nodes j form the next smallest class of Σ-entries in the lexicographic ordering. The Lyapunov equation simplifies to σ_ij = ζ∑_k ∈(j) m_jkσ_ki = ζ m_jiσ_ii = 2 ζ^2 m_ji which corresponds to the unique trek between i and j of length ℓ+r = 1. These two cases prove the initial induction hypothesis.

Fix any two (not necessarily distinct) vertices i, j and a non-empty (ℓ,r)-trek T between them. This trek decomposes into a unique (ℓ,r-1)-trek T_1 from i to l, for l ∈(j), and the edge l → j. It also decomposes uniquely into an (ℓ-1,r)-trek T_2 from k to j, for k ∈(i), together with the edge k → i. By (<ref>), the coefficient of m_T in σ_ij is thus ζ times the sum of the coefficients of m_T_1 in σ_il and m_T_2 in σ_kj. Using the induction hypothesis (since σ_il and σ_kj are both less than σ_ij) gives

    coefficient of m_T = ζ·(2 ζ^ℓ+rℓ+r-1ℓ + 2 ζ^ℓ+rℓ+r-1ℓ-1) = 2 ζ^ℓ+r+1ℓ+rℓ,

as claimed. This immediately proves the induction step when i ≠ j. In case i = j, there is also the empty trek from i to i to account for and its trek term arises as required from (<ref>) via C_ii = 2.







Consider the graph in <Ref>. Using the trek rule formulation for σ_ij from <ref>, we get 

    σ_11   =    2ζ, 
    σ_22   =    4ζ^3m_21^2 +2ζ, 
    σ_33   =    4ζ^3m_31^2 +2ζ, 
    ⋮
    σ_34   =    6ζ^4m_21m_31m_42 + 6ζ^4m_31^2m_43 + 2ζ^2m_43.

For instance, observe that there are two treks from the vertex 2 to itself, the empty trek (corresponding to the monomial 2ζ for σ_22) and the trek 2← 1 → 2 (4ζ^3m_21^2). Similarly, there are three treks between the vertices 3 and 4, which are the edge 3→ 4 (2ζ^2m_43), the trek 2← 1 → 3 → 4 (6ζ^4m_21m_31m_42), and 3← 1 → 3 → 4 (6ζ^4m_31^2m_43). For the last trek 3← 1 → 3 → 4, we have ℓ=1 and r=2, which gives us the coefficient 6ζ^4.



This trek rule is reminiscent of the original trek rule for linear structural equation models <cit.>. However, now each trek monomial is weighted by a binomial coefficient. While the trek rule we present here only parameterizes a proper subset of _ and is less general than that of <cit.>, it provides an explicit rational formula for the entries of Σ in terms of the parameters M which makes it significantly easier to construct explicit covariance matrices in _ with the desired dependence properties. We remark that the very recent preprint of <cit.> gives a generalization of <Ref>.



 §.§ Conditional Independence on Treks



We now turn to the case where the considered graph is a single trek, for which we show that the associated Lyapunov model does not satisfy any CI relations.
Before proceeding, we note that if  is a trek from node 1 to  with top node t, then  a drift matrix M∈ℝ^E() 
    
    has zero-pattern
    
    M = ([   d_1     *                              ;           ⋱     ⋱                        ;             d_t-1     *                  ;                     d_t                  ;                       * d_t+1            ;                             ⋱     ⋱      ;                                   *    d_;       ])

    with diagonal entries d_1,…,d_ and the symbol *
    indicating arbitrary values.  Such matrices are stable if and only
    if the diagonal entries are negative.
    





Let  be an (ℓ,r) trek between i and j. Let K = V() ∖{i, j} be the remaining vertices. Then i j | K in _. 


The proof of this proposition and its several supporting lemmas can be found in <ref>. The following example provides an explicit family of covariance matrices in the Lyapunov model for a small trek which exhibit the conditional dependence stated in <Ref>. The proof of the previous proposition generalizes this family of covariance matrices to larger treks and shows that the conditional dependence still holds by inductively computing certain entries in the adjoint matrix of Σ_iK, jK. The Schur complement expansion of |Σ_iK, jK| with respect to the principal minor Σ_K,K can then be used to show that |Σ_iK, jK| ≠ 0 by inductively computing the lowest degree term which appears in |Σ_iK, jK| when it is viewed as a polynomial in ζ. 






Let  be the trek as shown in <Ref>. Setting every edge parameter m_ji to 1 for all i→ j ∈ E(),

we get the following covariance matrix from the trek rule which lies in the model:

    Σ=[            2ζ          2ζ^2          2ζ^2          2ζ^3;          2ζ^2       4ζ^3+2ζ          4ζ^3          6ζ^4;          2ζ^2          4ζ^3       4ζ^3+2ζ     6ζ^4+2ζ^2;          2ζ^3          6ζ^4     6ζ^4+2ζ^2 12ζ^5+4ζ^3+2ζ ].

Observe that the submatrix Σ_{1,2,3},{2,3,4} has determinant -8ζ^6, which is non-zero for ζ≠0, implying that 2 4 |{1,3} in the model _.


We now introduce a technique to extend the previous proposition to all conditioning sets K ⊆ V() ∖{i, j}.  In Lyapunov models with fixed diffusion matrix C=2 _V, we cannot simply pass to marginal distributions as, in 

general, removing a row and corresponding column in a covariance matrix Σ∈_ does not yield

a distribution in the Lyapunov model over the shortened trek in which the removed node is skipped.

Instead, we provide a construction in which the random variables indexed by nodes in V() ∖ K become perfectly correlated with random variables indexed by elements of K.  This construction then allows us to argue that the counterexample from <ref> can be `lifted' to counterexamples on longer treks between i and j that contain additional nodes not in the conditioning set K.







To introduce the construction, we recall the following notation. 
For a node k for which (k) is a singleton set, we identify (k) with its single element, i.e., the unique node j with j→ k. Similarly,  we identify (k) with the unique node j such that k→ j.  Moreover, for k ∈, we define a map that shifts indices as

    ψ_-k: 
        ∖{k}→,    
        x ↦
            x    if  x < k; 
    
            x-1    if  x > k.


Now, consider a trek  between i and j with top node t and of length |V()|≥ 4. If |(t)| = 2, let c_l and c_r be the child nodes on the left and on the right side of the trek, respectively. Let k ∈ V()∖{i,j} be a node on the trek, and consider the subset of nodes V_-k = V()∖{k} obtained by removing k.  Define a shorter trek 

    _-k = (V_-k,E_-k)


by constructing the edge set E_-k as follows.
If k≠t,  replace the 2-path (k) → k →(k) by the single edge (k)→(k). If k=t, the trek  has two non-trivial sides as k∉{i,j}.

In this case, if the right side of  has length 1, we replace c_l ← k → c_r by c_l→ c_r. If k = t and the right side consists of more than one edge, we replace c_l ← k → c_r by c_l← c_r. The construction is depicted in <Ref>.
In all cases, the resulting graph is shorter by one node, as the k-th node is removed. Note that a similar definition is possible by adding the reverse edge c_l → c_r in the case where k = t and both sides of the trek consist of more than one edge. The distinction between the second and third case is necessary to keep the trek structure when removing a top node from a trek where one side consists of exactly one edge. Otherwise, the resulting graph could be a directed path, e.g., _-k could be i  c_l  c_r in <Ref>. 
In the case that the nodes V() = [] are numbered consecutively from left to right, we implicitly apply ψ_-k to the elements of V_-k and the start- and endpoint of each edge in E_-k so that we have V_-k = [-1] and E_-k⊆ [-1] × [-1].



Given a matrix Σ, we write 

    Σ_-k,-k = Σ_[]∖{k},[]∖{k}

for the submatrix obtained by removing the kth row and column of Σ.
Given a set of matrices ⊆^×, we denote its topological closure by .

 
    Let  be a trek between i and j containing more than 3 nodes.  Let t be the top node of , and let k ∉{i,t,j} be a further node on .  Then for every matrix Σ^* ∈__-k, there exists a matrix Σ∈_ such that 
    
    Σ_-k,-k = Σ_-(k),-(k) = Σ^*.




    Let ≥ 4 be the number of nodes on the trek .
    Without loss of generality, assume that the nodes on  are numbered consecutively from left to right, i.e., i = 1, j =, and k ∈{2,…,-1}∖{t}. 

    From Σ^* ∈__-k, we are to construct a suitable
    larger covariance matrix Σ∈_.  To do this,
    we make the
    new node at index k a copy of its parent on the
    trek by duplicating the respective row and column from Σ^*. In the resulting distribution,
    
    
    k and (k) are perfectly correlated. 
    Note that (k) = {k+1} for k < t, and (k)={k-1} for k > t. Thus, in the first case, the kth and k+1st row and column of Σ are duplicates of each other, while in the second case, the kth and k-1st row and column of Σ are duplicates of each other. The proof in both cases is analogous as the drift matrix of a single trek is defined in a symmetric way around its top node as seen in (<ref>). We only give the proof for the second case, so assume k > t.
    Furthermore, let U:={1,…,k-1},
    W:={k+1,…,}, and write ψ≡ψ_-k for
    the map from (<ref>). 
    Then, ψ(U) = U and ψ(W) = {k, …, -1}.
    
    We note that the case of a directed path, i.e., a trek with
    i=t=1, is included here as k > 1 by assumption.  

    Given Σ^* ∈__-k, there exists a stable matrix M^* ∈^(-1)×(-1) such that the Lyapunov equation 
    
    M^* Σ^* + Σ^* (M^*)^T + 2_-1 = 0

    is fulfilled.
    In the following, we show that our construction of inserting a
    duplicate of the row and column of Σ^* at index (k) in
     indeed yields a matrix Σ∈_.

    

    To achieve this, we construct a sequence of drift matrices M^(m) as follows. First, partition
    
    Σ^* = 
        ([      2*Σ^*_U 2*Σ^*_U,ψ(W);                          ; 2*Σ^*_ψ(W),U   2*Σ^*_ψ(W);                          ;              ])

    as well as 
    
    M^* = 
        ([      2*M^*_U          2*0;                          ; 2*M^*_ψ(W),U   2*M^*_ψ(W);                          ;              ]),
          where   
        M^*_ψ(W),U = 
        ([ 0    ⋯    0   m^*_k,k-1;           0     0 
    ⋮
     0;             ]).

    Considering (<ref>) block-wise results in the following four equations:
    
        
  * M^*_UΣ^*_U + Σ^*_U[-0]M^*_U + 2_k-1 = 0, 
        
  * M^*_UΣ^*_U,ψ(W) + Σ^*_U,ψ(W)M^*_ψ(W) + Σ^*_UM^*_ψ(W),U = 0, 
        
  * M^*_ψ(W)Σ^*_ψ(W) + Σ^*_ψ(W)M^*_ψ(W) + 2I_-k + M^*_ψ(W),UΣ^*_U,ψ(W) + Σ^*_ψ(W),UM^*_ψ(W),U = 0, 
        
  * M^*_ψ(W)Σ^*_ψ(W),U + Σ^*_ψ(W),U[-0]M^*_U + M^*_ψ(W),UΣ^*_U = 0, 
    
    where equations <ref> and <ref> are transposes of each other.
    Given the stable matrix M^*, these equations are uniquely fulfilled by the corresponding submatrices of Σ^*.
    
    Now, let m ∈_>0 and define
    
    M^(m) = 
        ([   2c|2*M^*_U          2*0        2c2*0;          2c|                        2c;            0            m           -m          2c0;       2c|2*0    m^*_k,k-1 2c2*M^*_ψ(W);          2c|            0           2c;              ]) ∈^×.

    Further, let
    
    Σ^(m) = 
        ([   2c|2*Σ^(m)_U    2*Σ^(m)_U,k  2c2*Σ^(m)_U,W;            2c|                            2c;   2c|Σ^(m)_k,U      Σ^(m)_k,k    2cΣ^(m)_k,W; 2c|2*Σ^(m)_W,U    2*Σ^(m)_W,k    2c2*Σ^(m)_W;            2c|                            2c;                ]) ∈^×

    be the solution of the resulting Lyapunov equation 
    
    M^(m)Σ^(m) + Σ^(m) (M^(m))^T + 2_ = 0

    induced by M^(m) for every m ∈_>0. 
    
    Due to the symmetry of the Lyapunov equation, any unique solution of (<ref>) is symmetric. Since M^(m) is stable, the solution Σ^(m) is unique and therefore symmetric, so we only have to solve six matrix equations that arise by block-matrix multiplication, denoted by letters in brackets as follows:
    
    M^(m)Σ^(m) + Σ^(m) (M^(m))^T + 2_ = 0 ≡([ 2c|2*(A)    2*(B)  2c2*(C);      2c|                2c;      2c|      (D)    2c(E);    2c|2*       2*  2c2*(F);      2c|                2c;          ]).


    The upper left block Σ^(m)_U of the partitioned matrix
    Σ^(m) is constant as it is the unique solution of
    equation <ref>. Proceeding recursively, the other
    blocks of Σ^(m) can be shown to converge for m →∞ as well, with limits given by the relevant parts of Σ^*. The details of these calculations can be found in <Ref>.
    Combining the results, we have
    
    lim_m →∞Σ^(m) = 
        ([      2c|2*Σ^*_U     2*Σ^*_U,(k)  2c2*Σ^*_U,ψ(W);             2c|                              2c;    2c|Σ^*_(k),U     Σ^*_(k),(k)  2cΣ^*_(k),ψ(W); 2c|2*Σ^*_ψ(W),U  2*Σ^*_ψ(W),(k)    2c2*Σ^*_ψ(W);             2c|                              2c;                 ]) ∈_,

as desired.


 
    Let  be a trek with endpoints i and j. Let ∅≠ K ⊊ V()∖{i,j} and let k ∉{i,j}∪ K be a node on the trek. If there is 
    
    Σ^* ∈__-k such that  | Σ^*_ψ_-k(iK),ψ_-k(jK)| ≠ 0,

    then there is 
    
    Σ∈_ such that  |Σ_iK,jK| ≠ 0.




    By continuity of the determinant, it suffices to construct a matrix Σ∈_ such that | Σ_iK,jK| ≠ 0.  Considering the closure of the model is crucial as it allows us to apply <ref> for the construction of Σ.   Then as k ∉{i,j}∪ K, we find that 
    
    | Σ_iK,jK| 
            = | Σ^*_ψ_-k(iK),ψ_-k(jK) |
             ≠ 0.






    
















































    

























 §.§ Conditional Independence in Zig-Zag Graphs




We now introduce the family of graphs that will be the key element needed to complete the proof of <ref>. 


Let _1, …, _p be treks such that for i ∈ [p-1], there exists a node k_i on the right branch of _i which is the endpoint of the left branch of _i+1. A zig-zag graph  (<Ref>) is a directed acyclic graph obtained by performing an iterated 1-clique sum of these treks along the nodes k_i, i.e.

    = _1 ⊕_k_1_2⊕_k_2⋯_p-1⊕_k_p-1_p.














Let  be a zig-zag graph with endpoints i and j, and let K ⊆ V() ∖{i, j} be a set such that for each collider k ∈ it holds that s ∈ K where s is the unique sink in (k). Then i j | K in the Lyapunov model associated to . 



To see this we use <Ref> inductively on the number p of treks that  is obtained from. If  is simply a trek then the result already holds by <ref>. So suppose that _p is a zig-zag graph with tails obtained from combining p treks and that for any set K' ⊆ V(_p) of the required form it holds that k j | K' where k and j are the endpoints of _p. Let  be a trek from i to s containing a node labelled k which is not on the same branch as i and let _p+1 be the 1-clique sum of  and _p at k. Recall that this is exactly how every zig-zag graph is produced as pictured in <Ref>. Note that if k and i are on the same branch then checking that i j | K on _p+1 naturally reduces to checking i j | K on a smaller zig-zag _p which immediately follows from the induction hypothesis. 


To complete the induction we must show that i j | K where K is of the form K = T ∪ L ∪ K' such that T ⊆ V() ∖{i}∪(k), s ∈ L ⊆(k) and K' ⊆_p is a set such that k j | K'. Lastly, let l ∈ L be the node closest to k. Then observe that after rearranging the rows and columns of Σ_iK, jK it has the block form

    Σ_iK, jK = 
    cccc    jK'                                  l T                              L ∖ l  
    l(ccc)
    l K'              Σ_lK', j K'   ∗   ∗
     
    i T                    0                                   Σ_iT, lT   ∗
    
    L ∖ l     Σ_L ∖ l, jK'   Σ_L ∖ l, lT   Σ_L ∖ l,L ∖ l
     .

Now observe that if we specialize this matrix by setting m_ab = 0 and m_la = 0 for all a,b ∈ L ∖ l then the above block matrix immediately simplifies to

    Σ_iK, jK = 
    cccc    jK'                                  l T                              L ∖ l  
    l(ccc)
    l K'              Σ_lK', j K'   ∗   ∗
     
    i T                    0                                   Σ_iT, lT   ∗
    
    L ∖ l      0                                    0                                   Σ_L ∖ l, L ∖ l
     .

Thus we see that for any such matrix obtained from the submodel corresponding to this specialization, |Σ_iK, jK| = |Σ_lK', j K'|  |Σ_iT, lT|  |Σ_L ∖ l, L ∖ l|. Now observe that |Σ_lK', j K'| ≠ 0 by our induction hypothesis since this corresponds to conditional dependence of a zig-zag with endpoints l and j which is composed of only p treks. Further observe that |Σ_iT, lT| ≠ 0 by <ref>. The last factor is a principal minor and thus is nonzero for any Σ∈__p+1. So any Σ which comes from the above submodel satisfies |Σ_iK, jK| ≠ 0 and thus i j | K for the Lyapunov model _. 
































Consider the zig-zag graph  pictured in <Ref>. Then 3 5 |{1, 4} since for any Σ∈_, it holds that

    Σ_{3,1,4}, {5, 1, 4} = 
    cccc    5                  4                 1 
    l(ccc)
    4      σ_45   σ_44   σ_14
     
    3       0                 σ_34   σ_13
    
    1       0                 σ_14   σ_11
     .

As was shown in the previous proof, this matrix is block upper-triangular with blocks σ_45 and Σ_{3,1}, {4,1}. 
Thus |Σ_{3,1,4}, {5, 1, 4}| = σ_45 |Σ_{3,1}, {4,1}| where the second term in the product is guaranteed to be nonzero for almost all choices of parameters by <ref>. 



We now have all the necessary tools needed to complete the proof of <ref>. 


If V ∖ ({i ,j }∪ K) separates i and j in the marginal independence graph  then <ref> immediately implies that i  j | K holds for all Σ∈_. It remains to show that if i  j | K then V ∖ ({i, j}∪ K) separates i in j in . We instead show the contrapositive: if i and j are connected given V ∖ ({i, j}∪ K) in , then there exists Σ∈_ such that i j | K. We will prove this by constructing a specific Σ using the trek rule described in <ref> and then showing that |Σ_iK, jK| ≠ 0 which by <Ref> implies that the conditional dependence i j | K holds.



If i and j are connected in  given V ∖ ({i ,j }∪ K) then there is path in  from i to j which uses only the vertices in {i, j}∪ K. Without loss of generality, this path is simple, i.e., it does not repeat a vertex. Let i = k_0, k_1, …, k_s, k_s+1 = j be the sequence of vertices on this path where {k_1, …, k_s}⊆ K. Each edge k_p  k_p+1 in  corresponds to a trek in . Even though the path in  is simple, these treks can still intersect in . Suppose that the edges k_p  k_p+1 and k_q  k_q+1 with p < q correspond to treks T_1 and T_2 in  which intersect. There are four cases depending on where this intersection occurs which are depicted in <Ref>. In the first three cases, there is a trek from k_p to k_q+1 through the intersection vertex — corresponding to a trek from i to j through m in <Ref>. This allows us to shorten the path in  by omitting k_p+q, …, k_q. The resulting path in  still has all its vertices contained in K.
The fourth case <Ref> goes from i over m to l, back up to m and then to j, omitting k from the path in . The walk in  traced out by this is a zig-zag with a tail m → l whose endpoint l is still in K.

For every pair of treks, multiple of these cases can occur simultaneously and each case can occur multiple times. One can freely choose any such intersection and resolve it according to <Ref> which results in a shortened path in . Iterating this procedure produces a simple path in  from i to j whose inner vertices are all contained in K. The union of the corresponding treks in  is a zig-zag with tails ⊆ as in <Ref>. 

By construction the subgraph  has the following property: for each collider m on , there is tail starting at m and completely contained in  whose end vertex is in K. Hence, <Ref> applies and shows that i j | K' on _ where K' is the set of vertices on  which are in K. Since  is a subgraph of , <Ref> implies that i j | K on _.








§ ACKNOWLEDGEMENTS


Tobias Boege was partially supported by the Academy of Finland grant number 323416 and by the Wallenberg Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation. Benjamin Hollering was supported by the Alexander von Humboldt Foundation. Mathias Drton, Pratik Misra, and Daniela Schkoda acknowledge support from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 883818). Sarah Lumpp acknowledges support by the Munich Data Science Institute (MDSI) at Technical University of Munich as part of the focus topic CAUSE. Daniela Schkoda and Sarah Lumpp were also supported by the DAAD programme Konrad Zuse Schools of Excellence in Artificial Intelligence, sponsored by the Federal Ministry of Education and Research.


Part of this research was performed while some of the authors were visiting the Institute for Mathematical and Statistical Innovation (IMSI), which is supported by the National Science Foundation (Grant No. DMS-1929348). Benjamin Hollering and Tobias Boege also thank the Combinatorial Coworkspace (funded by Combinatorial Synergies SPP 2458). 

We thank Shiva Darshan for helpful discussions regarding the proof of <Ref>.














    


































plain







§ PROOFS




 §.§ Proof of <ref>


In this subsection we provide a proof of <ref> by constructing a specific submodel of the Lyapunov model and then inductively showing that the determinant of the submatrix Σ_iK, jK does not vanish for a trek  with endpoints i and j where the conditioning set K = V() ∖{i, j}. 

Let  be a trek with endpoints i and j such that the length of the left branch, whose sink is i, is ℓ and the length of the right branch, whose sink is j, is r. Let K = V() ∖{i, j} be the set of all other nodes on . Let '_⊆_ of the Lyapunov model on  obtained from the trek rule in <ref> by setting every edge parameter m_qp = 1 for all p→ q ∈ E(). In other words

    '_ = {Σ∈_n  : σ_ij = ∑_T : (ℓ, r)-trek
    from i to j 2 ζ^ℓ+r+1ℓ+rℓ}.

This follows directly from <ref> since every trek monomial m_T = ∏_e ∈ Tm_e = 1. Thus we can think of any matrix Σ∈'_ as a matrix whose entries are univariate polynomials in ζ. Our goal is now to show that for almost all Σ∈'_, |Σ_iK, jK| ≠ 0. To do this, we will focus on computing the lowest degree term in |Σ_iK, jK| as a polynomial in ζ.
Formally, the lowest degree term of a univariate polynomial f = ∑_k=k_0^n f_k ζ^k with f_k_0≠ 0 is (f)  f_k_0ζ^k_0.

Before we begin, note that every entry of every Σ∈_' is divisible by 2ζ. In the following sequence of lemmas, we will instead consider a slight alteration of the above parameterization given by

    ”_  = {Σ∈_n  : σ_ij = ∑_T : (ℓ, r)-trek
    from i to jζ^ℓ+rℓ+rℓ}.

This simply corresponds to factoring out the common 2ζ from each entry of Σ which does not change whether or not a minor of Σ vanishes. Thus if Σ exhibits a conditional dependence then 2ζ Σ will exhibit the same conditional dependence. To prove that our original model _ satisfies i j | K, it suffices to prove it for a matrix in _”.

The following easy facts are used throughout the proofs in this section:


  * The lowest degree term is multiplicative: (fg) = (f) (g).

  * Since  is a trek, there is a unique shortest trek between any two vertices and thus there is a well-defined notion of distance on the trek.

  * Let the shortest trek between i and j have length (ℓ, r). Since it is unique, we have (σ_ij) = ℓ+rℓζ^ℓ+r for every Σ = Σ(ζ) ∈_”.

  * An important special case of this formula occurs when i and j are on the same branch of . In that case ℓ = 0 or r = 0 and the binomial coefficient is 1.

We begin with the following lemma concerning the lowest degree terms of principal minors. 


































Let  be a trek and Σ∈_”. For any K ⊆ V we have (|Σ_K|) = 1. 



Under the parametrization of _”, any diagonal entry σ_ii has (σ_ii) = 1 coming from the empty trek on i, while every off diagonal entry has ((σ_ij)) ≥ 1. The result then follows immediately from the Leibniz formula for the determinant. 


In the following, we compute minors of the form |Σ_iK,jK| which are decisive for conditional independence by <Ref>. If K = V ∖{i,j}, then this minor is, up to sign, the adjoint entry (Σ)_ij. To facilitate inductive proofs via Schur's formula (which computes |Σ_iK,jK| using entries of (Σ_K)), it is important to not only show that these minors are nonzero, but also to consistently compute the signs.

 
Throughout this section we assume the following numbering of the vertices V = {1,2,…,n} of the trek: the leaf on the left branch is labeled 1, its parent 2, and so on, until the top node is labeled ℓ+1; from there we descend down with ℓ+2 and so on until the leaf on the right branch is labeled n = ℓ+r+1. Any minor |Σ_I,J| is computed by ordering the rows and columns of the matrix with respect to this ordering.




Let  be a trek and Σ∈_”. Let i, j ∈ V() be distinct and (ℓ, r) be the length of the shortest trek between them. For any K ⊆ V() ∖{i,j}, either |Σ_iK,jK| = 0 or ((|Σ_iK,jK|)) ≥ℓ+r. If i and j are adjacent, then (|Σ_iK,jK|) = ζ.



Consider the Leibniz formula

    |Σ_iK,jK| = ∑_π: iK → jK 
    bijective(π) ∏_k ∈ iKσ_k π(k).

Note that Σ_iK,jK is a matrix with a particular ordering on its rows and columns. This ordering fixes a bijection between iK and jK with respect to which (π) is well-defined.

It suffices to show that each summand is either zero or has lowest term of degree≥ℓ+r. Fix a bijection π: iK → jK. The image π(i) either equals j or is an element k_1 ∈ K. In the latter case, we may apply π again to get π(k_1) = π^2(i) which either equals j or k_2 ∈ K. Continuing in this way, collect all iterated images of i into a set A ⊆ iK including i and excluding the j which is eventually reached. Observe that

    (∏_k ∈ Aσ_k π(k))
      = ∑_k ∈ A(σ_k π(k)) ≥(σ_ij) = ℓ + r,

since the sequence (σ_i π(i)), (σ_π(i)π^2(i)), …, (σ_π^m(i)j) given by A encodes an undirected path from i to j which is at least as long as the shortest trek.

The lower bound in (<ref>) can only be attained with equality if A is completely contained in the shortest trek ' connecting i and j and if the successive images π(i), π^2(i), … are ordered such that π(i) is the closest node to i, π^2(i) the second closest and so on; while all elements of K ∖ A are fixed points of π and do not contribute to the degree. In this case, the σ_kk, k ∈ K ∖ A, have lowest term 1 and the shortest treks between adjacent members of the sequence i, π(i), π^2(i), …, j give a disjoint decomposition of ' hence achieving degree exactly ℓ+r in total. If i and j are adjacent, then ℓ+r = 1 and the only bijection π which satisfies this sends i to j directly and fixes every other k ∈ K. Since i and j are adjacent, they appear in the same place in the ordering of iK and jK, respectively, hence π corresponds to the identity permutation of the rows and columns and thus we compute (|Σ_iK,jK|) = (σ_ij) = ζ.


As the previous proof shows, the lower degree bound ℓ+r is achievable and the Leibniz formula always has summands of this degree. Whether or not these summands cancel to cause a bump in lowest degree depends on the locations of i and j on the trek and to each other, as well as the conditioning set K.

 
In the following proofs, we will use the Schur complement expansion of the determinant:

    _K(i,j) |Σ_iK,jK| = σ_ij |Σ_K| - Σ_i,K(Σ_K) Σ_K,j = σ_ij |Σ_K| - ∑_p,q ∈ Kσ_ip_pqσ_qj,

where _pq = (Σ_K)_pq. To use Schur's formula to expand the determinant by the (i,j)-entry of Σ_iK,jK as indicated, we have to permute the ith row as well as the jth column to be the first row and column, respectively. The symbol _K(i,j) denotes the sign change in the determinant incurred by this. This sign depends only on the difference of the positions of i in iK and j in jK. We will assume below that K contains all interior nodes on the shortest trek ' between i and j. In this case and if i ≠ j, _K(i,j) can be computed as

    _K(i,j) = (-1)^n' = (-1)^|j-i|+1,

where n' is the number of nodes on ', which equals |j-i|+1 under <Ref>.




Let  be a trek and Σ∈_”. Let i, j ∈ V() be distinct, non-adjacent vertices on the same branch of . Denote the length of the shortest path ' between i and j by ℓ > 1. If K ⊆ V() ∖{i,j} contains all interior nodes on ', then either
|Σ_iK,jK| = 0 or (|Σ_iK,jK|) > ℓ.





Without loss of generality, we may suppose by symmetry that both i and j are on the left branch of the trek and i < j. Then the shortest path connecting them is also the unique shortest trek connecting them, which has length (ℓ, 0).

We perform induction on ℓ≥ 2. The case ℓ = 2 can be handled using the methods of <Ref>. The shortest trek ' between i and j is the form i → k → j where k ∈ K by assumption. Using the terminology of the proof of <Ref>, the only two bijections iK → jK which contribute to the ζ^2 term are:


  * π_1 sending i ↦ j and fixing every other element. Since k is between i and j in the ordering, the element i in iK occupies the same place as k in jK; and k in iK the same place as j in jK. Hence π_1 corresponds to an adjacent transposition of the rows and columns and has (π_1) = -1. This summand is thus -ζ^2.

  * π_2 sending i ↦ k ↦ j and fixing every other element. By similar reasoning as in the previous case, this corresponds to the identity permutation of the rows and columns and thus the contribution is ζ^2.

The two summands cancel and therefore either the entire determinant vanishes or the lowest term has degree > 2.

For the induction step, we use the Schur complement expansion (<ref>). By <ref> and the definition of _”, we know (σ_ij|Σ_K|) = ζ^ℓ. The goal is to show that for each choice of p, q ∈ K, either σ_ip_pqσ_qj = 0 or (σ_ip_pqσ_qj) ≥ℓ and in case of equality to compute the coefficient of the ζ^ℓ term.

We distinguish four cases (up to exchanging p and q or switching branches) depending on the locations of p and q with respect to i and j:


  * In case of <Ref>, notice that the shortest trek from i to p strictly includes the entire path from i to j, so ((σ_ip_pqσ_qj)) ≥((σ_ip)) > ℓ.

  * In case of <Ref>, the reason is similar. <Ref> shows ((σ_ip_pq)) > ℓ.

  * In case of <Ref>, we know by <Ref> that ((_pq)) > ℓ.

  *  The last case is depicted in <Ref>.

Then (σ_ip) = ζ^a and (σ_qj) = ζ^b and a+b+c=ℓ. So

    ((σ_ip_pqσ_qj)) = ℓ((_pq)) = c.

Notice that p and q are on the same branch of . If they coincide (c=0), then (_pq) = 1 by <Ref>; if they are adjacent (c=1), <Ref> yields _pq = -|Σ_K∖{q},K∖{p}| = -ζ. Otherwise, the induction hypothesis ensures _pq = 0 or (_pq) > c, which do not contribute to the lowest term. 

This implies that

    (∑_p,qσ_ip_pqσ_qj)    = ∑_p, q 
     c = 0(ζ^a · 1 ·ζ^b) + ∑_p, q 
     c = 1(ζ^a · (-ζ) ·ζ^b) 
       = ℓζ^ℓ - (ℓ - 1)ζ^ℓ = ζ^ℓ.


Thus the only case which contributes to the ζ^ℓ term in |Σ_iK,jK| is <ref>. Its contribution cancels with (σ_ij|Σ_K|) = ζ^ℓ in (<ref>), implying the result. 




Let  be a trek and i, j ∈ V() be nodes on different branches of  (in particular neither of them is the top node). Denote the length of the shortest path ' between i and j by (ℓ, r). If K ⊆ V() ∖{i,j} contains all interior nodes on ', then there exists Σ∈_” such that i j | K. 



Without loss of generality, we may assume that i < j in the ordering, so that i is on the left branch and j on the right. We claim that (|Σ_iK,jK|) = (-1)^ℓ+r+1ℓ+r-2ℓ-1ζ^ℓ+r which shows that |Σ_iK,jK| is non-zero and yields conditional dependence for all but a finite number of values of ζ.

We prove this statement by induction on ℓ, r ≥ 1, again using the Schur complement expansion (<ref>),

    (-1)^ℓ+r+1 |Σ_iK, jK| = σ_ij|Σ_K|  - Σ_i,K(Σ_K) Σ_K,j = σ_ij|Σ_K| - ∑_p,q ∈ Kσ_ip_pqσ_qj

with _pq = (Σ_K)_pq.

By <ref> and the definition of _”, we have (σ_ij |Σ_K|) = ℓ+rℓζ^ℓ+r.

The other part of the expansion is a sum over σ_ip_pqσ_qj which we deal with term by term, so fix p, q ∈ K. If p is strictly closer to j than q is (or, symmetrically, q is closer to i than p is), then (σ_ipσ_qj) > (σ_ipσ_pj) ≥(σ_ij) = ℓ+r. Additionally, by <Ref>, p and q must be contained in V('), i.e. i < p ≤ q < j in our ordering of vertices by <Ref>.

Hence, to find the lowest-order terms, we only need to consider

    p, q ∈ V(') ∖{i,j}⊆ K such that i < p ≤ q < j.


With these remarks, the basis of induction is easy to show. For ℓ = r = 1 the only possible option in (<ref>) is that p = q = top node of. Then (σ_ip_pqσ_qj) = ζ· 1 ·ζ = ζ^2. With (σ_ij |Σ_K|) = 2 ζ^2 we get (|Σ_iK,jK|) = (-1)^3 · (2ζ^2 - ζ^2) = -ζ^2, as required.



In the general case, the reduction via (<ref>) allows us to identify all relevant choices of p,q with decompositions of ' into three disjoint parts: the trek i to p, the trek p to q and the trek q to j.

There are three cases:



  * Suppose that p and q are both on the left branch of ' and suppose that they are not both equal to the top node. Then the trek decomposes into four paths as in <Ref>. We have (σ_ipσ_qj) = c+rcζ^a+c+d. By <Ref>, the lowest exponent of σ̂_pq in this case is b if and only if b = 0 or b = 1. Otherwise it is larger than b and a+b+c+d > ℓ+r disqualifying those cases from contributing to the lowest terms of the overall expression, so assume b = 0 or b = 1. Then (σ̂_pq) = (-1)^bζ^b by <Ref>. In total, this case contributes ∑_c = 1^ℓ-1c+rc when b=0 and -∑_c=0^ℓ-2c+rc when b=1 to the lowest-order terms. (The different summation ranges for c reflect that we must have a ≥ 1 and a+c = ℓ and also that if b = 0, in order for p = q not to be the top node, we must have c ≥ 1 whereas if b = 1, we only need c ≥ 0.) Together, this gives ℓ+r-1ℓ-1-1. The case where p and q are both on the right branch is analogous and contributes ℓ+r-1r-1 = ℓ+r-1ℓ-1.


  * Suppose that p is on the left branch and q is on the right branch but neither of them is the top node. Then the trek decomposes into four paths as in <Ref> and (σ_ipσ_qj) = ζ^a+d. Here the induction hypothesis applies and computes |Σ_pL,qL| = (-1)^q-p+1b+c-2b-1ζ^b+c, where L = K ∖{p,q}. (Note that L contains all interior nodes on the shortest trek between p and q since K had the same property for i and j.) However, there is another sign to account for coming from the cofactor: _pq = (Σ_K)_pq = (-1)^q-p |Σ_pL,qL| = -b+c-2b-1ζ^b+c. The sign only depends on q-p, again because L contains all vertices between them in the ordering.

In total, this case contributes -∑_b=1^ℓ-1∑_c=1^r-1b+c-2b-1 = 1 - ℓ+r-2ℓ-1 to the lowest term.


  * There is a special case when p and q both coincide with the top node of the trek, as in <Ref>. In this case, they are both simultaneously on the left and the right branch but this case was not counted in the previous bullet points. As in the base case of the induction, the contribution is (σ_ip_pqσ_qj) = ζ^ℓ+r.

In total,

    (-1)^ℓ+r+1(|Σ_iK,jK|) 
       = ( ℓ+rℓ - ℓ+r-1ℓ-1 - ℓ+r-1ℓ_= 0 + 2 
     - (1 - ℓ+r-2ℓ-1) - 1 ) ζ^ℓ+r
       = ℓ+r-2ℓ-1ζ^ℓ+r,

as claimed.


The last remaining case needed to complete the proof of <ref> is the case when  is a trek such that ℓ = 0 or r = 0, meaning that  is actually a path. In this case <ref> shows that the lowest degree term can have degree strictly higher than ℓ + r. For this reason, we instead analyze the highest degree term in ζ which appears in the expansion of |Σ_iK, jK|. Let (f) denote the highest degree non-zero term of a univariate polynomial f.



Let  be a directed path from i to j and K=V()∖{i,j} be the remaining vertices. Then there exists Σ∈_” such that i j | K. 


To show that there exist Σ∈_” for which i j | K, it is enough to show that |Σ_iK,jK| as a univariate polynomial in ζ has a non-zero coefficient.

To this end, we compute the determinant of (Σ_iK,jK), which is the matrix obtained by taking the highest degree term of each entry of Σ_iK,jK, at ζ=1. If this value is non-zero, it follows that |(Σ_iK,jK)| ≠ 0 is the highest degree term of |Σ_iK,jK| which is thus non-zero as well.

We claim that |(Σ_iK,jK)|=1 at ζ=1.
Recall that _” is defined by the parametrization which maps σ_ij to the sum of ζ^ℓ+rℓ+rℓ over all possible treks between i and j. Hence, the entries of (Σ_iK,jK) only take into account the longest treks between nodes (with the highest value of ℓ+r).

For simplicity, we number the vertices from 1 to n with i=1 and j=n. Now, the longest and unique trek between 1 and k for any k≤ n is precisely the directed path from 1 to k. However, the longest trek between a and k for any 1<a<k is one where the left and right branches are directed paths from 1 to a and 1 to k, respectively. This implies that the term with highest degree for σ_1k has coefficient 1 (as ℓ=0) and for that of σ_ak has coefficient a+k-2a-1. This gives us that the (a,b) entry of (Σ_{1,2,3,…, n-1},{2,3,…,n}) at ζ=1 is precisely a+b-2a-1. However, using the identity in <cit.>, we can obtain an LU decomposition of this matrix where the diagonal terms of both L and U are 1. This implies that |(Σ_iK,jK)|=1 at ζ=1, which proves our claim. 




 §.§ Proof Details for <ref>




    In the proof of <ref>, we solve the following six equations resulting from the block-matrix partition for the corresponding blocks of Σ^(m) for m →∞:
    
        
  (A) M^*_UΣ^(m)_U + Σ^(m)_U[-0]M^*_U + 2_k-1 = 0 

        This equation is uniquely solved by Σ^(m)_U = Σ^*_U due to <ref>.
        
        
  (B) M^*_UΣ^(m)_U,k + mΣ^(m)_U,k-1 -m Σ^(m)_U,k = 0 

        The matrix M^*_U - m_k-1 is invertible due to the structure of M^*, so <ref> together with (A) yields
        
    lim_m →∞Σ^(m)_U,k = lim_m →∞Σ^(m)_U,k-1 = Σ^*_U,k-1.

        
        
  (C) M^*_UΣ^(m)_U,W + Σ^(m)_U,WM^*_ψ(W) + Σ^(m)_U,k(m^*_k,k-1, 0, …, 0) = 0 

        Taking the limit of the left-hand side for m →∞ yields
        
    M^*_UΣ^(m)_U,W + Σ^(m)_U,WM^*_ψ(W) + Σ^(m)_U,k (m^*_k,k-1, 0, …,0)  
        M^*_U( lim_m →∞Σ^(m)_U,W) + ( lim_m →∞Σ^(m)_U,W) M^*_ψ(W) + Σ^*_U,k-1 (m^*_k,k-1, 0, …,0),

        where we employed (B) in the last summand. This last summand can further be rewritten as 
        
    Σ^*_U,k-1 (m^*_k,k-1, 0,…,0) = Σ^*_UM^*_ψ(W),U

        by attaching a (k-2) × (k-1) zero matrix to the existing row vector and thereby extending it to M^*_ψ(W),U. Setting the limit to zero yields equation <ref> which is solved by Σ^*_U,ψ(W). This gives us lim_m →∞Σ^(m)_U,W = Σ^*_U,ψ(W).
        
        
  (D) 2-2mΣ^(m)_k,k+m(Σ^(m)_k,k-1+Σ^(m)_k-1,k) = 0 

        Solving for Σ^(m)_k,k, letting m go to infinity, and using the result from (B) yields 
        
    lim_m →∞Σ^(m)_k,k 
                = lim_m →∞(1/m + Σ^(m)_k,k-1+Σ^(m)_k-1,k/2) 
                = Σ^*_k-1,k-1+Σ^*_k-1,k-1/2 
                = Σ^*_k-1,k-1.

        
        
  (E) m Σ^(m)_k-1,W - m Σ^(m)_k,W+ Σ^(m)_k,k (m^*_k,k-1, 0,…,0) + Σ^(m)_k,WM^*_ψ(W) = 0 

        The matrix M^*_ψ(W) - m_-k is invertible due to the structure of M^*, so the transposed version of <ref> together with (C) and (D) yields
        
    lim_m →∞Σ^(m)_k,W = lim_m →∞Σ^(m)_k-1,W = Σ^*_k-1,ψ(W).

        
        
  (F) M^*_ψ(W)Σ^(m)_W + Σ^(m)_WM^*_ψ(W) + 2_-k

        + (m^*_k,k-1, 0,…,0)^T Σ^(m)_k,W + Σ^(m)_W,k(m^*_k,k-1,0,…,0) = 0 

        We use that Σ^(m)_k,W→Σ^*_k-1,ψ(W) as well as Σ^(m)_W,k→Σ^*_ψ(W),k-1 for m →∞ due to (E) and take the limit on both sides. This yields for the left-hand side
        
    M^*_ψ(W)Σ^(m)_W + Σ^(m)_WM^*_ψ(W) + 2_-k
            + (m^*_k,k-1, 0,…,0)^T Σ^(m)_k,W + Σ^(m)_W,k(m^*_k,k-1,0,…,0) 
        M^*_ψ(W)(lim_m →∞Σ^(m)_W) + (lim_m →∞Σ^(m)_W) M^*_ψ(W) + 2_-k
            + (m^*_k,k-1, 0,…,0)^T Σ^*_k-1,ψ(W) + Σ^*_ψ(W),k-1(m^*_k,k-1,0,…,0) 
    
                =      M^*_ψ(W)(lim_m →∞Σ^(m)_W) + (lim_m →∞Σ^(m)_W) M^*_ψ(W) + 2_-k
            + M^*_ψ(W),UΣ^*_U,ψ(W) + Σ^*_ψ(W),UM^*_ψ(W),U,

        where we employed the same technique of rewriting the last two summands as for (C). Setting the limit to zero yields equation <ref> that is solved by Σ^*_ψ(W). We conclude lim_m →∞Σ^(m)_W = Σ^*_ψ(W).
     


 
    Let A^(m) and C^(m) be convergent sequences of × l matrices, and let D,E,F ∈^× be constant matrices such that E - m _ is invertible for all m ∈_> 0.
    If a further matrix sequence B^(m) satisfies that 
    
    m · A^(m) - m · B^(m) + D · A^(m) + E · B^(m) + F · C^(m) =0

    for all m ∈_> 0, then
    
    lim_m→∞ B^(m) = lim_m→∞ A^(m).




    By collecting the terms and dividing by m, we have 
    
    (1/m D + _) A^(m) + (1/m E - _) B^(m) +  1/m F · C^(m) = 0.

    Since (1/m E - _) is invertible, we can write
    
    B^(m) = - (1/m E - _)^-1((1/m D + _) A^(m) - 1/m F · C^(m)).

    As matrix inversion is continuous and A^(m) and C^(m) are convergent, letting m go to infinity on both sides yields the result.





 Paper Title: Facilitating heterogeneous effect estimation via statistically efficient categorical modifiers.





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    Facilitating heterogeneous effect estimation via statistically efficient categorical modifiers
    
    
  
  
  
  
    Daniel R. KowalAssociate Professor, Department of Statistics and Data Science, Cornell University and Department of Statistics, Rice University (mailto:dan.kowal@cornell.edudan.kowal@cornell.edu). 
    Research  was sponsored by the National Institute of Environmental Health Sciences (R01ES028819) and the National Science Foundation (SES-2214726). The findings and conclusions in this publication are those of the author(s) and do not necessarily represent the views of the NIH, the U.S. government, or the North Carolina Department of Health and Human Services, Division of Public Health.
    
    
    August 5, 2024
=============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================

 



§ ABSTRACT 
Categorical covariates such as race, sex, or group are ubiquitous in regression analysis. While main-only (or ANCOVA) linear models are predominant, cat-modified linear models that include categorical-continuous or categorical-categorical interactions are increasingly important and allow heterogeneous, group-specific effects. However, with standard approaches, the addition of cat-modifiers fundamentally alters the estimates and interpretations of the main effects, often inflates their standard errors, and introduces significant concerns about group (e.g., racial) biases. We advocate an alternative parametrization and estimation scheme using abundance-based constraints (ABCs). ABCs induce a model parametrization that is both interpretable and equitable. Crucially, we show that with ABCs, the addition of cat-modifiers 1) leaves main effect estimates unchanged and 2) enhances their statistical power, under reasonable conditions. Thus, analysts can, and arguably should include cat-modifiers in linear regression models to discover potential heterogeneous effects—without compromising estimation, inference, and interpretability for the main effects. Using simulated data, we verify these invariance properties for estimation and inference and showcase the capabilities of ABCs to increase statistical power. We apply these tools to study demographic heterogeneities among the effects of social and environmental factors on STEM educational outcomes for children in North Carolina. 0An  package  is available. 


Keywords:  Discrete data; Interactions; Penalized Estimation; Regression analysis


1.45 


§ INTRODUCTION


Interactions are remarkably valuable  in linear regression analysis. 

In particular, interactions between a categorical (or nominal) variable and either a continuous or categorical variable—referred to here as cat-modifiers—are crucial for discovering and quantifying heterogeneous effects. A prominent example is  race: due to structural racism and discrimination, the effects of many important variables on health and life outcomes vary by race  
<cit.>,  
with race often interacting with sex or socioeconomic status <cit.>. 
Cat-modifiers are also highly relevant for studying gene-environment interactions <cit.> and appear broadly in the social and behavioral sciences <cit.>. Within statistics, the urgency of cat-modifiers is perhaps best known by Simpson's paradox <cit.>, where the omission of cat-modifiers produces entirely misleading associations.  
 




 
 
 
 
 
 






 


Yet there are significant obstacles to the inclusion of cat-modifiers in linear regression analysis. Broadly, cat-modifiers alter the interpretation of the main effects, introduce concerns about equity across categorical groups (e.g., for race, sex, and other protected groups), 
change the main effect estimates, and typically inflate the main effect standard errors (SEs). Consequently, cat-modifiers are often omitted or misreported <cit.>, which falsely suppresses heterogeneity. 

We argue that, with the right parametrization,  cat-modifiers can readily, and arguably should be included in linear  regression models with categorical covariates. 
To establish ideas, suppose we have p continuous covariates x = (x_1,…, x_p)^⊤ and K categorical variables C = (C_1,…, C_K)^⊤ with L_k levels for each categorical variable k=1,…,K. We consider regression models for data {( x_i,  c_i, y_i)}_i=1^n parameterized by a linear regression function μ( x,  c) which typically models the conditional expectation 𝔼(Y | x,  c) or a transformed version for generalized linear models. We distinguish between two classes of linear models: those that do not include cat-modifiers and those that do.  First, the main-only model includes multiple continuous and categorical variables, but no interactions:

    μ^M( x,  c) = α_0^M +  x^⊤α^M + ∑_k=1^K β_k, c_k^M

or in Wilkinson notation, . Second, the cat-modified model expands (<ref>) to allow categorical-continuous and categorical-categorical interactions:

    μ( x,  c) = α_0 +  x^⊤α + ∑_k=1^K β_k, c_k +  ∑_k=1^K  x^⊤γ_k, c_k + ∑_k=1^K-1∑_k'=k+1^K γ_k,k',c_k, c_k'

or equivalently, , using pairwise interactions for convenience. Our notation emphasizes that the parameters in (<ref>) and (<ref>) are fundamentally distinct, even though these models are nested. 

The advantage of the cat-modified model is the ability to estimate heterogeneous, group-specific effects for each x_j. While both models specify group-specific intercepts (consider (<ref>) and (<ref>) with x = 0), 

only the cat-modified model features group-specific slopes: 

    μ_x_j'( c) μ(x_j+1,  x_-j,  c) - μ(x_j,  x_-j,  c) = α_j + ∑_k=1^K γ_j, k, c_k.

By comparison, the slopes in the main-only model do not depend on c:  μ_x_j^M' μ^M(x_j+1,  x_-j,  c) - μ^M(x_j,  x_-j,  c) = α_j^M. 





For concreteness, we consider two popular cases. Empirical examples are given in Tables <ref> and <ref>, respectively, and these cases are revisited subsequently. 
[ANCOVA]
    Suppose we have p=1 continuous variable x ∈ℝ and K=1 categorical variable  with L_R groups. The main-only model (<ref>) is then 
    
    μ^M(x, r) = α_0^M + xα_1^M + β_r^M

    or equivalently, , with group-specific intercepts, μ^M(0, r) = α_0^M + β_r^M for each  group r, but a global (race-invariant) slope, μ_x^M' =  α_1^M. Thus, (<ref>) produces parallel lines with race-specific vertical shifts. By comparison, the cat-modified model (<ref>) is 
    
    μ(x, r) = α_0 + x α_1 + β_r + x γ_r

    or equivalently, , with group-specific intercepts μ(0, r) = α_0 + β_r and group-specific slopes  μ_x'(r) = α_1 + γ_r for each  group r. 


[Two-way ANOVA]
    Suppose we have K=2 categorical variables  and  with L_R and L_S groups, respectively. The main-only  model (<ref>) is then
    
    μ^M(r,s) = α_0^M + β_1,r^M + β_2,s^M

    or equivalently, , while the cat-modified model (<ref>) is 
    
    μ(r, s) = α_0 + β_1,r + β_2,s + γ_rs

or equivalently, . 



The central challenge is that expanding from the main-only model to the cat-modified model  alters the interpretations, estimates, and inference for the main effects, i.e., the parameters {α_0^M, α^M, β_k,c_k^M} in (<ref>) or the analogous terms {α_0, α, β_k,c_k} in (<ref>). 
If these impacts are detrimental, then a quantitative modeler may be reluctant to include cat-modifiers. The key determinant is the model parametrization or identification strategy used for the categorical variable coefficients. Specifically, both models (<ref>) and (<ref>) require additional constraints to interpret and estimate the model parameters:  the main-only intercepts {α_0^M, β_k,c_k^M} are overparametrized, while the cat-modified intercepts {α_0, β_k,c_k, γ_k,k', c_k,c_k'} and slopes {α, γ_k,c_k} are overparametrized. The identifications determine the interpretations of all main and interaction parameters and the statistical properties of their estimators.

The most popular identification strategies are  problematic for cat-modified models. Reference group encoding (RGE)  is the overwhelming default, including for all major software implementations of  generalized linear regression (, SAS, Python,  MATLAB, Stata, etc.).  With RGE, a reference group is selected for each categorical variable C_k and removed: β_k,1^M = 0 for all k in (<ref>) and β_k,1 = 0, γ_k,1 =  0, γ_k,k', 1,c_k' = γ_k,k', c_k, 1 =0 for all (c_k,c_k') in (<ref>) (using 1 for each reference group without loss of generality). This is equivalent to using L_k -1 “dummy variables" to encode each C_k. 

Despite the simplicity of RGE, the implied notion of “main effects" in (<ref>) significantly impedes the use of cat-modifiers.  For the main-only model, the jth main effect is a global slope, α_j^M = μ_x_j^M', invariant of c; yet for the cat-modified model, RGE fixes γ_j,k,1 =  0 for all j so that α_j = μ_x_j'( 1) is the group-specific x_j-effect with all categorical variables set to their reference groups (c =  1). 



First, this main effect parametrization is statistically inefficient: SEs for α̂_j are typically larger than those for α̂_j^M—the intersection of all reference groups is a subset of the data with a much smaller effective sample size—while the group-specific x_j-effect μ_x_j'( c) may be smaller for the reference groups (c=  1) than for other groups or globally (i.e., α_j^M). We illustrate this effect in Table <ref>: with RGE, the main effects for the cat-modified model are attenuated and sacrifice power compared to those for the main-only model (see also Sections <ref> and <ref>). Similar effects occur for categorical-categorical interactions (see Table <ref>). Of course, these parameters refer to different functionals of μ( x,  c); yet crucially,  they are presented identically as “main effects" in statistical software output and manuscript tables  <cit.>. In fact, fewer than half of recent social science publications even reported the reference category <cit.>. This leads to misleading conclusions about effect magnitudes, directions, and heterogeneity <cit.>. 















Second, RGE is inequitable: the main effect elevates a single (reference) group above the others. In Table <ref>,  White is the reference group: the main effect α_1 = μ_x'() is  the x-effect for the  White group, while the interaction effect γ_r = μ_x'(r) - μ_x'() is the difference between the x-effect for race r and that for the White group. RGE presents the reference groups (main effects) as “normal" while all the other groups (interaction effects) are “deviations from normal"—almost always without any explicit labeling. This framing biases the interpretations of results <cit.>. 

The problem is compounded for regularized regression: when coefficient estimates are regularized toward zero, the group-specific slopes are statistically biased toward the reference group slope (γ_r → 0 implies μ_x'(r) →μ_x'()). Beyond the obvious inequities—including (racial, gender, etc.) bias in the estimators—this shrinkage obscures potential differences between the x-effects for dominant (e.g., White, Male, etc.) and nondominant groups. 
Thus, RGE undermines progress toward statistical methods that  promote equity <cit.>. 


Finally, RGE is difficult to interpret: each main effect and interaction in (<ref>) must be traced back to all reference groups. Consider Example <ref> (and Table <ref>), using White and Male for the reference groups:  the main effects in the cat-modified model (<ref>) are  α_0 = μ(, ), β_1,r = μ(r, ) - μ(, ) for each race r, and β_2,s = μ(, s) - μ(, ) for each sex s. Each main effect is anchored at both reference groups, which then affects the interpretations of the interaction effects γ_rs. These challenges are accentuated with multiple categorical covariates and interactions as in (<ref>). 










An alternative identification strategy uses sum-to-zero constraints (STZ). STZ identifies the parameters by restricting the group-specific coefficients to sum to zero:  ∑_ℓ=1^L_kβ_k,ℓ^M = 0 for all k in the main-only model and ∑_ℓ=1^L_kβ_k,ℓ = 0, ∑_ℓ=1^L_kγ_j,k,ℓ = 0 for j=1,…, p, and  ∑_ℓ=1^L_kγ_k,k', ℓ,c_k' = 0 and ∑_ℓ=1^L_kγ_k,k', c_k,ℓ =0 for all (c_k, c_k') in the cat-modified model. STZ is common for ANOVA models <cit.> and has been incorporated into regularized regression <cit.>. STZ eliminates the need for a reference group, and thus resolves the inequities of RGE. However, STZ does not offer any special statistical properties for estimation of (<ref>) or (<ref>), nor does it establish a clear connection between the “main effects" in (<ref>) and (<ref>). As a result, it is difficult to interpret the parameters under STZ, while the addition of cat-modifiers may have unpredictable or detrimental effects on the main effect estimates and inferences (see Section <ref>). 







To address these limitations, we advocate and analyze   abundance-based constraints (ABCs) for identification and estimation with cat-modified models. Broadly, ABCs identify parameters using group abundances (see Section <ref>). 
ABCs are sufficiently general and may be combined with ordinary least squares (OLS), maximum likelihood, and modern regularized estimation techniques. 
The benefits are summarized by “EEI":

    
  * Efficiency: ABCs permit the inclusion of cat-modifiers 1) without altering the main effect OLS estimates and 2) either maintaining or increasing their statistical power, under reasonable conditions; 
    
  * Equity: ABCs do not require a reference group and thus eliminate the alarming inequities under default approaches (RGE); and 
    
  * Interpretability: main effects  are identified as group-averaged  parameters, interaction effects are group-specific deviations from these group-averaged parameters, and both sets of parameters inherit meaningful notions of sparsity. 

ABCs effectively remove the impediments to cat-modifiers, thus facilitating richer regression analyses of heterogeneous effects. Of course, ABCs cannot guarantee that cat-modifiers will be practically or statistically significant, especially when the effective sample sizes for interactions are small. Rather, with ABCs, there is virtually nothing to lose by expanding from the main-only model (<ref>) to the cat-modified model (<ref>); yet the potential gains include greater statistical power for the main effects and discovery of  heterogeneous effects. 



We emphasize that ABCs, in various forms and by other names, have deep historical roots, but have lacked sufficient motivation to encourage widespread adoption.  <cit.> and <cit.> considered identification strategies for ANOVA models based on arbitrary group-specific weights. 
Ultimately, both adopted STZ. <cit.> suggested ABCs for the simple ANCOVA (<ref>) so that the estimated intercept  would  equal the sample mean  (see also Theorem <ref>). However, there was no consideration of cat-modifiers or multiple covariates and no case made for any of EEI. Among nonlinear models, <cit.> and <cit.> used an ABC-like approach to avoid estimating main effects, instead focusing exclusively on interactions to optimize individual treatment rules. More subtly, they required independence between the cat-modifier (treatment) and any modified covariates, which is not usually satisfied for observational data and not required for our results. For the two-way ANOVA (<ref>), <cit.> briefly mentioned ABCs only to dismiss them, claiming  they “complicate the interpretation of the model parameters and make it difficult to fit the model...especially when other covariates are present." Here, we forcefully argue the opposite, embodied by EEI—each of which applies 

with multiple covariates present. 
ABCs are considered concurrently in  <cit.>, which focuses on issues of equity with race as a single cat-modifier.  


Contrasts provide an alternative perspective on identification of  (<ref>) and (<ref>): dummy coding, effects coding, and weighted effects coding (WEC) are respectively linked to RGE, STZ, and ABCs. Although WEC has garnered recent support <cit.>,
this work did not consider general cat-modifiers or any EEI. Instead, WEC has been mainly limited to two-way ANOVAs (<ref>) or (<ref>) and only advocated in restrictive settings with 
“certain types of unbalanced data that are missing not at random” <cit.> or “categories of different sizes, and if these differences are considered relevant”  <cit.>. Our case is much broader and more direct: ABCs are ideal to identify coefficients on any categorical variables and  cat-modifiers should be included in many, if not all linear models. Further, we enforce ABCs using linearly-constrained optimization, which—unlike contrasts—is well-suited for regularized regression (see Section <ref>).





Lastly, we acknowledge additional perspectives on the cat-modified model (<ref>). A widely-used approach is subgroup analysis, which subsets the data into groups (for all combinations of c) and then fits separate regression models (e.g., ). The appeal is that it estimates group-specific slopes without the complicated interpretations of the parameters in (<ref>) under default approaches (RGE). However, subgroup analysis does not provide  estimates or inference for the main effects, cannot incorporate regularization or borrow information across groups, and does not allow direct testing for interaction effects. Notably, ABCs offer the same (and more) benefits without any of these drawbacks. 
Related, <cit.> advocated for marginal means. These quantities, like group-specific slopes and fitted values, will be identical for all (minimally sufficient) identification strategies under maximum likelihood estimation. Thus, it does not distinguish among identification strategies. 
However, the identification strategy remains key for 1) parameter interpretation, estimation, and inference and 2) regularized regression and variable selection. 



  















 




The paper is organized as follows. We introduce ABCs  in Section <ref>, both for parameter identification and statistical estimation. Our main results on theory for estimation and inference with ABCs are in Section <ref>. Simulation studies are in Section <ref> and a real data example is in Section <ref>. We conclude in Section <ref>. Supplementary material includes proofs of all results, details for generalized linear models, additional simulation results, and supporting data information and analysis.  
0An  package  is available. 













 






















    





 



 

 


 



 





§ IDENTIFICATION, ESTIMATION, AND INFERENCE WITH ABCS
 







The goal of ABCs is to enforce model identifiability while maintaining EEI. We first describe the model parametrization and interpretation, and then show how to compute regularized regression estimators and inference  using linearly-constrained optimization. The main properties for estimation and inference are in Section <ref>.



 §.§ Parameter identification with ABCs


For motivation, consider (<ref>) from Example <ref>: identifiability is obtained by constraining ∑_r=1^L_Rπ_r β_r = 0 and ∑_r=1^L_Rπ_r γ_r = 0 for some chosen nonnegative weights {π_r}_r=1^L_R. RGE sets π_1 = 1 and π_r = 0 for r>0, while STZ sets all π_r = 1. Instead, suppose we view each constraint as an expectation: 𝔼_π(β_R) = 0 and 𝔼_π(γ_R) = 0, where  R is a categorical random variable with probabilities {π_r}_r=1^L_R. Now, 

the main x-effect α_1 is equivalently the average of the group-specific slopes: 𝔼_π{μ_x'(R)} = 𝔼_π(α_1 + γ_R) =α_1. Of course, this notion of “average"—as well as the  accompanying properties for statistical estimation (Section <ref>)—depends entirely on the supplied probabilities {π_r}. 
 ABCs adopt a natural choice: the (population or sample) abundances by group. For instance, in 
Table <ref>, the ABCs specify
(π_, π_, π_) = (0.587, 0.351, 0.062) using  sample proportions. 

More broadly, we define ABCs for the cat-modified model (<ref>); special cases such as  (<ref>) simply omit the constraints for the omitted parameters. We express the ABCs in terms of π̂, which is the joint proportions across all categorical variables C = (C_1,…, C_K)^⊤ in the data { c_i}_i=1^n. ABCs may be defined using population or sample proportions; we prefer the latter because they are always available  and estimation properties are tractable and favorable (Section <ref>). First, ABCs for categorical main effects and categorical-continuous interactions are

    𝔼_π̂ (β_k, C_k)    = 0,    k=1,…, K 
    𝔼_π̂ (γ_j,k, C_k)    = 0,    k=1,…, K,    j=1,…,p.

Equivalently, ABCs may be expressed marginally and with summations: ∑_ℓ = 1^L_kπ̂_k, ℓβ_k, ℓ = 0, where {π̂_k,ℓ}_ℓ=1^L_k are the sample proportions for each categorical variable C_k, k=1,…, K,  and then similarly for each {γ_j,k,ℓ}_ℓ=1^L_k. The key implication is that, while the cat-modified model (<ref>) incorporates heterogeneity via mutual, group-specific slopes (<ref>),  ABCs  concisely identify each main x_j-effect as the average of this group-specific slope:

    α_j = 𝔼_π̂{μ_x_j'( C)}.

ABCs parameterize each main x_j-effect  by aggregating the group-specific slopes (<ref>), each weighted by its respective abundance in the data. Unlike RGE, ABCs do not elevate any single (reference) group, and thus avoid the accompanying inequities. 


The identification in (<ref>) also guides interpretation of the group-specific slope parameters {γ_j,k,ℓ}_ℓ=1^L_k. Consider (<ref>) from Example <ref>:  γ_r = μ_x'(r) - 𝔼_π{μ_x'(R)} is the difference between the group-specific slope for group r and the group-averaged slope. 
For the general cat-modified model (<ref>), isolating γ_j,k,c_k requires averaging over the remaining categorical variables C_-k with joint proportions π̂_-k with C_k=c_k  fixed: γ_j,k,c_k = 𝔼_π̂_-k{μ_x_j'( C_-k, c_k)} - 𝔼_π̂{μ_x_j'( C)}. 

Further simplifications are often available, since these averages only must include the categorical variables that act as cat-modifiers for x_j. In contrast with RGE, these group-specific coefficients are parameterized relative to a global main effect term (<ref>), rather than a single (reference) group (White, Male, etc.).





For categorical-categorical interactions, ABCs identify {γ_k,k', c_k, c_k'} by requiring

    𝔼_π̂_C_k | C_k'=ℓ(γ_k,k', C_k, ℓ)    = 0,   ℓ=1,…,L_k'
    𝔼_π̂_C_k'| C_k=ℓ(γ_k,k', ℓ, C_k')    = 0,   ℓ=1,…,L_k

for all (C_k,C_k') interactions based on the conditional proportions for categorical variable C_k given that the interacting  variable C_k' belongs to group ℓ (and vice versa). In conjunction,  (<ref>) and (<ref>) constitute ABCs. We illustrate (<ref>) using model  (<ref>) from Example <ref>: 
 𝔼_π̂_S | R=r(γ_rS) = 0 for  r=1,…,L_R and 𝔼_π̂_R | S=s(γ_Rs) = 0 for s=1,…,L_S, where π̂_S | R=r = {π̂_rs/π_r}_s=1^L_S is the conditional probability for each  given =r (similarly for π̂_R | S=s). Equivalently, (<ref>) may be expressed using the joint proportions π̂: for Example <ref>, this is ∑_s=1^L_Sπ̂_rsγ_rs = 0 for r=1,…,L_R and  ∑_r=1^L_Rπ̂_rsγ_rs = 0 for  s=1,…,L_S. Thus, all ABCs (<ref>) and (<ref>) can be written in terms of the joint probabilities π̂.  










 
 
 
 
 
 




There are several compelling reasons to identify the categorical-categorical interactions with (<ref>).  First,  it guarantees a global, group-averaged identification for the intercept: 


    Under ABCs, 
    the intercept parameter in  (<ref>) satisfies 𝔼_π̂{μ( 0,  C)} = α_0.

ABCs produce clean expressions and simple interpretations for these main effects: while cat-modifiers induce group-specific intercepts and slopes, ABCs identify suitably global, group-averaged quantities α_0 = 𝔼_π̂{μ( 0,  C)} and  α_j = 𝔼_π̂{μ_x_j'( C)} for j=1,…,p. This cannot occur for RGE and only occurs for STZ if the probabilities π̂ are exactly uniform. 
Second, (<ref>) orthogonalizes the main and interaction categorical effects: in fact, the OLS estimates of the main categorical effects {β_k, c_k} are identical between models that do (<ref>) or do not (<ref>) include cat-modifiers (Theorem <ref>). Finally, (<ref>) offers the interesting result that, if we were to instead combine the interacted covariates (C_k, C_k') into a single categorical variable (e.g., race-sex) with L_kL_k' levels, the main effect ABCs (<ref>) would be satisfied for this new categorical variable. Of course, doing so would sacrifice the ability to estimate the main effects {β_k, c_k}, but this internal consistency is reassuring. 

















 







 
 

















 
 
For implementation, it is sufficient to enforce L_k + L_k' -1 of the L_k + L_k' constraints in (<ref>). The choice of omitted constraint is arbitrary, since all constraints (<ref>) hold regardless:
 
    Suppose we apply (<ref>) to all but one interaction term: 
    𝔼_π̂_C_k | C_k'=ℓ(γ_k,k', C_k, ℓ) = 0 for ℓ=1,…,L_k' and 
    𝔼_π̂_C_k'| C_k=ℓ(γ_k,k', ℓ, C_k') = 0 for  ℓ=2,…,L_k. Then the same constraint holds for ℓ=1: 𝔼_π̂_C_k'| C_k=1(γ_k,k', 1, C_k') = 0. 
    




Finally, we emphasize that ABCs (<ref>) and (<ref>) are designed for parameter identification in the general cat-modified model (<ref>), which may be featured in generalized linear models (see the supplementary material, Section <ref>) and includes numerous important special cases, such as  main-only models (<ref>), ANCOVA models (Example <ref>), and two-way ANOVA models (Example <ref>), among many others.

 



 §.§ Estimation, inference, and sparsity with ABCs


ABCs are linear constraints and thus readily compatible with regularized regression. First, we consolidate the cat-modified model (<ref>) into a traditional regression structure: (μ( x_1,  c_1), …, μ( x_n,  c_n))^⊤ =  Xθ, where  X is the n × P matrix that includes an intercept, all (centered) continuous covariates, indicator variables for all levels of each categorical variable, and all specified interactions, and θ include all unknown regression coefficients. In the presence of at least one categorical covariate, X is rank deficient, say ( X) = P-m. We represent all ABCs (<ref>) and (<ref>) generically as A_π̂θ =  0, where is the m × P matrix of constraints with (  A_π̂) = m. 
Then, for a loss function ℒ( y,  X θ) for data y = (y_1,…,y_n)^⊤ and a coefficient penalty 𝒫(θ), we aim to solve 

    θ̂(λ) = min_θℒ( y,  X θ)  + λ𝒫(θ)     A_π̂θ =  0

and λ≥ 0 is a tuning parameter. We primarily focus on squared error loss  ℒ( y,  X θ) = ‖ y -   X θ‖^2 and either unpenalized estimation (λ=0) or (group) lasso and ridge regression with λ selected by cross-validation. 

One way to solve (<ref>) is to reparametrize to an unconstrained space with only P-m parameters. 

Let A_π̂^⊤ =  Q  R be the QR-decomposition 
with  columnwise partitioning of the P × P orthogonal matrix Q = ( Q_1:m :  Q_π̂ ) and similarly, R^⊤ = ( R_1:m, 1:m :  0). 
By construction, A_π̂ Q_π̂ =  0, so that for any (P-m)-dimensional vector θ_Q, the vector θ =  Q_π̂θ_Q satisfies  A_π̂θ =  0. Then, letting X_Q  X   Q_π̂,  (<ref>) is equivalently

    θ̂(λ) =  Q_π̂θ̂_Q(λ),   θ̂_Q(λ) = min_θℒ( y,  X_Q θ_Q) + λ𝒫( Q_π̂θ_Q).

Regularized regression with ABCs simply requires 1) computing the QR decomposition of A_π̂^⊤ and 2)  solving an unconstrained regularized regression problem. 



















 
















When ℒ is a negative log-likelihood,  θ̂_Q θ̂_Q(0) is a maximum likelihood estimator (MLE) and so is  θ̂. Hence, usual properties for MLEs apply to estimators with ABCs. Under standard regularity conditions,  (<ref>) satisfies 
 
    √(n)(θ̂ - θ) d→ N_P( 0,  Q_πℐ(θ_Q)^-1 Q_π^⊤) 
where ℐ is the Fisher information associated with θ_Q and π is the joint population probabilities for the categorical covariates C. Thus, it is straightforward to construct confidence intervals and conduct hypothesis tests for the coefficients θ.  When the model errors y_i - μ( x_i,  c_i) are Gaussian, uncorrelated, and homoskedastic, 



the OLS estimator under ABCs satisfies 

θ̂∼ N_P{θ, σ^2  Q_π̂( X_Q ^⊤ X_Q)^-1 Q_π̂^⊤}, even in finite samples. Although this distribution does not account for the sampling variability in π̂, this is typically quite small relative to the variability in θ̂. Our empirical analyses suggest that no further adjustments are needed (see Section <ref>). 



Finally, we emphasize the unique challenges of regularization and selection  for cat-modified models. Selection of interaction effects has primarily focused on high-dimensional, continuous-continuous interactions <cit.>. For cat-modified models with RGE,  coefficient  shrinkage introduces (racial, gender, etc.) biases: γ_j,k,c_k→ 0 implies μ_x_j'( c) →μ_x_j'( 1), so group-specific effects are pulled toward those for the reference (White, Male, etc.) groups.  With ABCs, no such biases occur: γ_j,k,c_k→ 0 implies μ_x_j'( c) →𝔼_π̂{μ_x_j'( C)} collapses to the group-averaged x_j-effect, which produces a reasonable notion of parameter sparsity.  

When λ > 0, it is possible to omit constraints and still obtain unique estimators. However, these estimators do not target identifiable parameters and thus are difficult to interpret. For lasso estimation, <cit.> observes that such “overparametrized" estimation tends to reproduce RGE by implicitly selecting a reference group, and thus inherits the same limitations as RGE. 
















§ THEORY FOR ESTIMATION AND INFERENCE WITH ABCS

A central nuisance with interactions is that they change the main effect estimates and SEs. Here, we show that ABCs circumvent these challenges for cat-modifiers. The main point is that, with ABCs, the addition of cat-modifiers is either 1) harmless, since it has little to no impact on main effects estimates and inference, or 2) beneficial, since it can reveal heterogeneity and improve statistical power for the main effects. 




 

 

 






  §.§.§ QUESTION: how do we interpret β_r (or β_s)?

Under model (<ref>) and the constraints (<ref>), it is difficult to isolate β_r. Usually the best bet is to consider μ( 0, R,S) and compute various expectations over the categorical covariates, like with α_0 in Lemma <ref>.

First, we know that 𝔼_π̂_S(β_S) = 0, so it might be reasonable to compute the the intercept averaged over all but race,

    𝔼_π̂_-R{μ( 0, r,S)}   = 𝔼_π̂_S{μ( 0, r,S)} = ∑_s=1^L_Sπ̂_s(α_0 + β_r + β_s + γ_rs) 
       = α_0 + β_r + ∑_s=1^L_Sπ̂_s γ_rs.

This operation eliminates β_s, but ∑_s=1^L_Sπ̂_s γ_rs is not zero in general. Note that this approach would yield α_0 + β_r for any main-only model (i.e., with γ_rs=0, even with additional categorical main effects). 



Instead, we might try to compute the conditional expectation of the intercept, which averages the intercept over sex for a given race:

    𝔼_π̂_S | R=r{μ( 0, R,S)}   = ∑_s=1^L_Sπ̂_rs/π̂_r(α_0 + β_r + β_s + γ_rs) 
       = α_0 + β_r + ∑_s=1^L_Sπ̂_rs/π̂_rβ_s.

Here, the interaction term γ_rs is eliminated, but a term with β_s remains. 

Idea 1: It is possible to modify the ABCs (<ref>) such that 𝔼_π̂_S | R=r(γ_RS) = 0 is replaced with 𝔼_π̂_S | R=r(β_S + γ_RS) = 0, similarly for each interaction term, so that the above conditional expectation would simplify: 𝔼_π̂_S | R=r{μ( 0, R,S)} = α_0 + β_r. This alternative maintains identifiability and Lemma 1, but sacrifices the estimation invariance in Theorem 1.  



Idea 2: Since the OLS estimator of β_r is the same whether or not the interaction is included, then perhaps the first approach is promising. In particular, for a model without the interaction, we can show that 

    β_r = 𝔼_π̂_-R{μ( 0, r,S)} - 𝔼_π̂{μ( 0, R,S)} = 𝔼_π̂_-R{μ( 0, r,S)} - α_0.

This offers at least some path toward an interpretation. However, it is not true for β_r when the interaction term γ_rs is present in the model. 





Possible Idea 3: just mention it as a race effect without much explanation...



 






§ SPECIAL CASES AND UNIFYING THEMES


Notation:
Consider categorical variables R (race), S (sex), and E (education level) with L_R, L_S, and L_E levels, respectively. For notation, we let β_R the coefficient for race and race group R, β_S the coefficient for sex and sex group S, etc. This notation is slightly overloaded, but context is clear. Similarly,  β_RS^* is the race-sex interaction for race R and sex S, and so forth. Let 𝔼_π̂_R denote averaging with respect to race proportions, e.g., 

    𝔼_π̂_R(β_R) = ∑_r=1^L_Rπ̂_r β_r.

where π̂_r is the proportion in race group r. For models without categorical-categorical interactions, ABCs simply imposed 𝔼_π̂_R(β_R) for each categorical variable (e.g., one for R, one for S, one for E). Now, further consideration is required. 

Let 𝔼_π̂ be a joint expectation over the mutual proportions of (R, S, E), and 𝔼_π̂_-R is the joint expectation over all but race, so that R=r is fixed. Finally, let  refer to a placeholder for (one or more) continuous covariates. The constraints below also apply for categorical-continuous interactions with , but instead on the slope terms. Thus, we focus on the intercepts to avoid redundancy. 



Goal: define ABCs to handle various categorical-categorical interaction combinations. Arguably, the main effects are  most interpretable if we can obtain constraints such that the intercept term μ( 0, R, S, E) satisfies

    𝔼_π̂{μ( 0, R, S, E)} = α_0 
    𝔼_π̂_-S{μ( 0, R, s,E)} = α_0 + β_s,    s=1,…,L_S 
    𝔼_π̂_-R{μ( 0, r, S,E)} = α_0 + β_r,    r=1,…,L_R 
    𝔼_π̂_-E{μ( 0, R, S,e)} = α_0 + β_e,    e=1,…,L_E

The first constraint gives α_0 a suitably global interpretation, and also implies that α̂_0 = y̅ (under OLS with ABCs), which is simple and invariant across many models. The latter constraints allow us to interpret the main effects, such as β_r, as the difference between the global average and the average over all but race, and specifically for race r. 

Note: previously also considered replacing 𝔼_π̂_-R with the conditional version, 𝔼_π̂_·| r. However, some of the estimates seemed nonintuitive and backwards...






  §.§.§ Main-only models


    :      μ( 0,R) = α_0 + β_R 
         
    𝔼_π̂_R (β_R) = 0
    
    
         𝔼_π̂{μ( 0,R)} = α_0





    :       μ( 0,R, S) = α_0 + β_R + β_S 
         𝔼_π̂_R (β_R) = 0 
    𝔼_π̂_S (β_S) = 0
    
         𝔼_π̂{μ( 0, R, S)} = α_0 
    𝔼_π̂_-R{μ( 0, r, S)} = α_0 + β_r,    r=1,…,L_R 
    𝔼_π̂_-S{μ( 0, R, s)} = α_0 + β_s,    s=1,…,L_S





    :      μ( 0,R,S, E) = α_0 + β_R + β_S + β_E
         𝔼_π̂_R (β_R) = 0 
    𝔼_π̂_S (β_S) = 0 
    𝔼_π̂_E (β_E) = 0
    
         𝔼_π̂{μ( 0, R, S, E)} = α_0 
    𝔼_π̂_-S{μ( 0, R, s,E)} = α_0 + β_s,    s=1,…,L_S 
    𝔼_π̂_-R{μ( 0, r, S,E)} = α_0 + β_r,    r=1,…,L_R 
    𝔼_π̂_-E{μ( 0, R, S,e)} = α_0 + β_e,    e=1,…,L_E

For connections with later specifications, note that these ABCs are equivalent to the following constraints (and yield the same implications):

    𝔼_π̂_-R(β_S + β_E) = 0 
    𝔼_π̂_-S(β_R + β_E) = 0 
    𝔼_π̂_-E(β_R + β_S) = 0

To see this, observe that the first two constraints here imply  
𝔼_π̂_S(β_S) =  -𝔼_π̂_E(β_E) and   𝔼_π̂_R(β_R) =  -𝔼_π̂_E(β_E), so 𝔼_π̂_S(β_S) = 𝔼_π̂_R(β_R). Applying the third constraint, we have     𝔼_π̂_R(β_R) = 𝔼_π̂_S(β_S) = 𝔼_π̂_E(β_E) = 0 as before. 

Note: in both cases, the joint (intercept) constraint is implied but not enforced directly.



  §.§.§ Cat-cat interaction models


    :      μ( 0,R,S) = α_0 + β_R + β_S + β_RS^* 
         𝔼_π̂ (β_R + β_S + β_RS^*) = 0 
    𝔼_π̂_S (β_S + β_rS^*) = 0,    r=1,…,L_R 
    𝔼_π̂_R (β_R + β_Rs^*) = 0,    s=1,…,L_S
    
         𝔼_π̂{μ( 0, R, S)} = α_0 
    𝔼_π̂_-R{μ( 0, r, S)} = α_0 + β_r,    r=1,…,L_R 
    𝔼_π̂_-S{μ( 0, R, s)} = α_0 + β_s,    s=1,…,L_S

This includes the correct number of constraints (1 + L_R + L_S) and the implications are consistent with those in the main-only setting.








    :      μ( 0,R,S, E) = α_0 + β_R + β_S + β_E + β_RS^*
         𝔼_π̂ (β_R + β_S +β_E + β_RS^*) = 0 
    𝔼_π̂_S (β_S + β_rS^*) = 0,    r=1,…,L_R 
    𝔼_π̂_R (β_R + β_Rs^*) = 0,    s=1,…,L_S 
    𝔼_π̂_E (β_E) = 0
    
         𝔼_π̂{μ( 0, R, S, E)} = α_0 
    𝔼_π̂_-S{μ( 0, R, s,E)} = α_0 + β_s,    s=1,…,L_S 
    𝔼_π̂_-R{μ( 0, r, S,E)} = α_0 + β_r,    r=1,…,L_R 
    𝔼_π̂_-E{μ( 0, R, S,e)} = α_0 + β_e,    e=1,…,L_E


Here, we have 1 + L_R + L_S + L_E + L_RL_S terms and only L_RL_S + L_E -1 identified parameters, and thus require 2 + L_R + L_S constraints. These ABCs are equivalent to

    𝔼_π̂ (β_R + β_S +β_E + β_RS^*) = 0 
    𝔼_π̂_-R (β_S + β_E + β_rS^*) = 0,    r=1,…,L_R 
    𝔼_π̂_-S (β_R + β_E + β_Rs^*) = 0,    s=1,…,L_S 
    𝔼_π̂_-E (β_R + β_S + β_RS^*) = 0

To see this, observe that the first and last constraints imply 𝔼_π̂_E (β_E) = 0. Thus, we have added a zero to the middle terms and replaced the last constraint with an  equivalent one. 

 




    Under the ABCs  𝔼_π̂_R (β_R + β_Rs^*) = 0 for all s=1,…,L_S and  𝔼_π̂_S (β_S + β_rS^*) = 0 for all r=1,…,L_R, it follows that
    𝔼_π̂_R(β_R) = 𝔼_π̂_S(β_S).


    
    Let 𝔼_π̂_R (β_Rs^*) = k_1 and 𝔼_π̂_S (β_rS^*) = k_2 for all levels s,r. It suffices to show that k_1 = k_2. Rewriting as summations, the first ABC is k_1 = ∑_r π̂_r β_rs^* and therefore β_1s^* = π̂_1^-1(k_1 - ∑_r > 1π̂_r β_rs^*), using r=1 without loss of generality. Substituting into the second ABC, we have k_2 = ∑_s π̂_s β_1s^* 
        =  ∑_s π̂_s {π̂_1^-1(k_1 - ∑_r > 1π̂_r β_rs^*)}. Rearranging terms, this equality holds whenever 
        
    π̂_1 k_2    = k_1 -∑_r> 1π̂_r ∑_s π̂_s β_rs^* 
       = k_1 - ∑_r > 1π̂_r k_2 
       = k_1 - (1 - π̂_1)k_2,

        i.e., k_1 = k_2. 

These results hold for any quantities—parameters or estimators—that satisfy ABCs.  
By construction, we also have 𝔼_π̂_R(β_R) = 𝔼_π̂_S(β_S) = -1/2𝔼_π̂ (β_RS^*). This quantity need not be zero.


    : 
    μ( 0,R,S, E)    = α_0 + β_R + β_S + β_E + β_RS^* + β_RE^*
         𝔼_π̂ (β_R + β_S + β_E + β_RS^* + β_RE^*) = 0 
    𝔼_π̂_S (β_S + β_rS^*) = 0,    r=1,…,L_R 
    𝔼_π̂_E(β_E + β_rE^*) = 0,    r=1,…,L_R 
    𝔼_π̂_-S (β_R + β_E + β_Rs^* + β_RE^*) = 0,    s=1,…,L_S
    𝔼_π̂_-E(β_R + β_S + β_RS^* +  β_Re^*) =0,    e=1,…,L_E
    
         𝔼_π̂{μ( 0, R, S, E)} = α_0 
    𝔼_π̂_-R{μ( 0, r, S,E)} = α_0 + β_r,    r=1,…,L_R 
    𝔼_π̂_-S{μ( 0, R, s, E)} = α_0 + β_s,    s=1,…,L_S 
    𝔼_π̂_-E{μ( 0, R, S, e)} = α_0 + β_e,    e= 1,…, L_E

Here, we have 1 + L_R + L_S + L_E + L_RL_S + L_RL_E terms and only L_RL_S + L_RL_E - L_R identified parameters, and thus require 1 + 2 L_R + L_S + L_E constraints. If we add more terms interacted with race, then the 2 is replaced with the number of race-categorical interactions, and the number of levels for the additional categorical variables appears in the summation (like L_S and L_E). If another additive categorical variable is added, then just one additional constraint  is needed. 

The general formula here seems to be the following: apply the Implications for all terms except those categorical variables C interacted with R, and for those we constrain 𝔼_C(β_C + β_rC^*)=0. 

Alternative:

    𝔼_π̂ (β_R + β_S + β_E + β_RS^* + β_RE^*) = 0 
    𝔼_π̂_-R (β_S + β_E + β_rS^* + β_rE^*) = 0,    r=1,…,L_R 
    𝔼_π̂_E(β_rE^*) = 0,    r=1,…,L_R 
    𝔼_π̂_-S (β_R + β_E + β_Rs^* + β_RE^*) = 0,    s=1,…,L_S
    𝔼_π̂_-E(β_R + β_S + β_RS^* +  β_Re^*) =0,    e=1,…,L_E










    :  
       μ( 0,R,S, E) = α_0 + β_R + β_S + β_E + β_RS^* + β_RE^* + β_SE^*


Here, we have 1 + L_R + L_S + L_E + L_RL_S + L_RL_E + L_SL_E terms and only 1 + L_RL_S + L_RL_E + L_SL_E - L_R - L_S - L_E identified parameters, and thus require 2L_R +  2 L_S + 2L_E constraints.   



    𝔼_π̂_-R (β_S + β_E + β_rS^* + β_SE^*) = 0,    r=1,…,L_R 
    𝔼_π̂_-R (β_rE^*) = 0,    r=1,…,L_R 
    𝔼_π̂_-S (β_R +  β_E + β_Rs^* + β_RE^*) = 0,    s=1,…,L_S 
    𝔼_π̂_-S (β_sE^*) = 0,    s=1,…,L_S 
    𝔼_π̂_-E (β_R + β_S +β_RS^* + β_Re^*) = 0,    e=1,…,L_E 
    𝔼_π̂_-E (β_Se^*) = 0,    e=1,…,L_E 
    
         
        fixme

Note 1: we know this has too many constraints, at least by one. If we can show that it has too many constraint by two, then we can replace each joint version with L_R-1, L_S-1, and L_E-1 constraints, and then add the usual global constraint... 


Note 2: does it matter which joint term we “leave out"? Does the interpretation of the resulting coefficients change??





 






 








 §.§ Estimation invariance with ABCs

We establish conditions under which main effect OLS estimates are invariant to the addition of cat-modifiers under ABCs. These results make  minimal  assumptions about the true data-generating process and do not apply for other identifications (RGE, STZ, etc.). 

First, consider OLS estimation of the intercept. For an enormous class of linear models—with arbitrarily many continuous covariates, categorical covariates, and categorical-categorical interactions—ABCs ensure that the OLS-estimated intercept is always exactly equal to the sample mean, α̂_0 = y̅ n^-1∑_i=1^n y_i.

For any linear model of the form (<ref>) with 1) centered continuous covariates (x̅ =  0),  2) no categorical-continuous interactions (all γ_k,c_k =  0), and 3) ABCs (<ref>) and (<ref>), 
the OLS estimate of the intercept is α̂_0 = y̅. 

Simple models such as  yield the same intercept estimate as more complex models like .  This reaffirms the global interpretation of the intercept: under ABCs, α̂_0 targets the global intercept α_0 = 𝔼_π̂{μ( 0,  C)} (Lemma <ref>). Of course, y̅ is a good estimator for the marginal expectation of Y, so  α̂_0 is appropriately global—even in the presence of categorical variables and their interactions. For models with at least one categorical variable, this result cannot occur for any other identification (RGE, STZ, etc.).  Theorem <ref> extends   <cit.> to allow for categorical-categorical interactions and arbitrarily many continuous covariates. 

Next, consider the impact of adding categorical-categorical interactions on estimation of the main effects. For concreteness, we consider a two-way ANOVA (Example <ref>).

    Under ABCs (<ref>) and (<ref>), the OLS estimates of all main effects are identical under the main-only model (<ref>) and the cat-modified model (<ref>): α̂_0^M = α̂_0, β̂_1,r^M = β̂_1,r for all r=1,…,L_R, and β̂_2,s^M = β̂_2,s for all s=1,…,L_S.

This estimation invariance applies to all (1 + L_R + L_S) main effects in (<ref>). Implicitly, we assume that the OLS estimates exist and are unique (i.e., empty categories are not permitted), but otherwise there are no requirements on the data-generating process. In particular, there are no assumptions of independence or uncorrelateness between the categorical covariates and no assumptions about their relationship with Y.  ABCs deliver a natural interpretation of the categorical variable coefficients: the main effects are deviations from the global mean (Theorem <ref>), while the interaction effects are deviations from the main effects in the main-only model (Theorem <ref>).  This result validates our choice of ABCs (<ref>) and especially (<ref>). Again, such invariance does not occur for other identifications.




Finally, we consider the addition of categorical-continuous interactions to main-only models. For clarity, we focus on a single categorical variable (K=1) with levels r=1,…, L_R, but showcase these principles empirically with multiple categorical variables (Section <ref>).  Following Example <ref>, we begin with a single continuous covariate x (p=1). Let σ̂_x[r]^2  n_r^-1 s_x[r]^2 - x̅_r^2 be the (scaled) sample variance of {x_i}_i=1^n for each group r, where n_r = nπ̂_r, s_x[r]^2 = ∑_r_i=r x_i^2 and x̅_r = n_r^-1∑_r_i=r x_i. If the continuous covariate has  the same scale for each group, then the OLS estimate of the coefficient on x is the same whether or not the cat-modifier is included.

    Under ABCs (<ref>) and the equal-variance condition
    
    σ̂_x[r]^2 = σ̂_x[1]^2    r=1,…,L_R,

    the OLS estimates for the main-only model (<ref>) and the cat-modified model (<ref>) satisfy  estimation invariance, α̂_1^M = α̂_1.

We apply Theorem <ref> as an approximation, α̂_1 ≈α̂_1^M whenever σ̂_x[r]^2 ≈σ̂_x[1]^2 for all r, which is reasonably robust to deviations from  (<ref>) (see Table <ref> and Section <ref>). This condition makes no requirements on the true associations between Y and (x,r) and generally allows the distribution of x to vary by r—as long as the scale is approximately constant. In particular, strong dependencies between x and r are permissible.

It is clarifying to consider violations of the equal-variance condition (<ref>), so that x varies substantially for some groups, but varies little for others. This scenario does not invalidate estimation with ABCs; rather, it decouples the coefficients on x (i.e., the main x-effects) from the models that do (<ref>) or do not (<ref>) include a cat-modifier. Arguably, the main-only model (<ref>) is no longer appropriate in this setting. The x-effect in (<ref>) is α_1^M = μ^M(x+1) - μ^M(x), which considers a one-unit change in x regardless of the group r. But when (<ref>) is violated, the scale of x—and a “one-unit change in x"—is no longer comparable across  groups. Thus,  group-specific slopes μ(x+1, r) - μ(x, r) = α_1 + γ_r are appropriate, which mandates the cat-modified model. As the global x-effect α_1^M from the main-only model is no longer appealing, the cat-modified model with ABCs instead identifies a global x-effect via the group-averaged quantity α_1 = 𝔼_π̂{μ(x+1, R) - μ(x, R)} as in (<ref>). In this setting, distinctness between α_1 and α_1^M is appropriate.

This result can be extended  for p continuous covariates, each of which is cat-modified:  . Here, the equal-variance condition (<ref>)  instead uses the (scaled) sample covariance between x_j and x_h in group r, _r( x_j, x_h)  n_r^-1∑_r_i = r (x_ij - x̅_j)(x_ih - x̅_h). 


    
    Consider the main-only model (<ref>) and the cat-modified model 
    (<ref>), each with K=1 categorical variable. 
    Under  ABCs (<ref>) and the equal-covariance condition 
    _r( x_j, x_h) = _1( x_j, x_h)
    
    for all r=1,…,L_R and each j,h=1,…,p, 
    the OLS estimates satisfy α̂ = α̂^M. 

Theorem <ref> ensures estimation invariance for all p continuous main effects, each of which is cat-modified. Thus, the equal-covariance condition is stricter than (<ref>). As with Theorem <ref>, we apply Theorem <ref> as an approximation, so that α̂≈α̂^M when equal-covariance approximately holds. 

Lastly, we establish a middle ground: , which is a cat-modified model with p continuous covariates and K=1 categorical variable, but now only x_1 is cat-modified. Instead of covariances between all pairs of covariates, the equal-variance condition now involves only x_1 and the residuals ê_1 from regressing x_1 on all other variables, . 


    Consider the main-only model (<ref>) with K=1 and the  cat-modified model 
    (<ref>) with K=1 and interactions only with x_1 (fix γ_j,r = 0 for all j > 1 and r=1,…,L_R). 
    Under ABCs (<ref>)  and the equal-variance condition 
    _r( x_1, ê_1) = _1( x_1, ê_1)
    for all r=1,…,L_R, the OLS estimates satisfy α̂_1^M = α̂_1. 
    
    
    
    

To understand this modified equal-variance condition, we can equivalently express _r(ê_1, x_1) = n_r^-1∑_r_i = r (x_i1^2 - x_i1x̂_i1) = σ̂_x_1[r]^2 - _r(x̂_1, x_1), where x̂_1 are the fitted values from . Theorem <ref> requires that the variability in x_1 explained by the remaining (continuous and categorical) covariates is the same within each group. When this condition is violated, a one-unit change in x_1 holding all else equal among x_2,…,x_p is no longer comparable across groups r. As with Theorem <ref>, the main-only model x-effect α_1^M is no longer appropriate; group-specific x_1-effects μ_x_1'(r) = α_1 + γ_1,r are preferred; and ABCs offer a substitute for the global slope parameter via the group-averaged x_1-effect, α_1 = 𝔼_π̂{μ_x_1'(R)}. 

















 








    
    












 §.§ Powerful inference with ABCs

A primary reason for the unpopularity of cat-modifiers is the loss of statistical power for the main effects. With RGE, cat-modifiers relegate the main effects to a single reference group, which shrinks the effective sample size and often attenuates global effects. Thus,  quantitative modelers may be reluctant to include cat-modifiers for fear of larger p-values, wider confidence intervals, and less power to identify important effects. Consequently, potential race-, sex-, or other group-specific effects may remain hidden. 




ABCs directly and uniquely address this challenge. With the addition of cat-modifier effects, we show that ABCs may actually reduce SEs for the main effects. The magnitude of this reduction increases with the effect size of the cat-modifier. Crucially, when the cat-modifier effect is unnecessary, then the main effect SEs match, but do not inflate, those for a (correct) main-only model. 



Consider two nested models, a main-only model and a cat-modified model. Our general result is that the cat-modified model with ABCs has smaller SEs for the main effects whenever the estimated residual variance is smaller for the cat-modified model,

    Ŝ^2 ≤Ŝ_M^2.

For the maximum likelihood estimators  Ŝ^2 =  ‖ê‖^2/n and Ŝ_M^2 =  ‖ê_M ‖^2/n, where ê and ê_M are the residuals from the cat-modified and main-only models,  respectively, (<ref>) is guaranteed: ‖ê‖^2 ≤‖ê_M ‖^2, typically with strict inequality. More commonly, the unbiased estimators  Ŝ^2 =  ‖ê‖^2/(n - d_M - d) and Ŝ_M^2 =  ‖ê_M ‖^2/(n - d_M) are used, where d_M+d and d_M are the number of identified parameters for the cat-modified and main-only models, respectively. In that case, (<ref>) 
requires that the adjusted-R^2 for the cat-modified model exceeds that for the main-only model, or equivalently,  
(‖ê_M ‖^2  - ‖ê‖^2) /‖ê_M ‖^2 ≥  d/(n-d_M), 

so that the (guaranteed) reduction in sum-squared-residuals from main-only to cat-modified must be  large enough to justify the addition of d parameters. This requirement is modest:  adjusted-R^2 is well-known to prefer overparametrized models, and thus (<ref>) is likely to hold even when the cat-modifiers are extraneous (see Section <ref>). When cat-modifiers are indeed necessary, the reduction from Ŝ_M^2 to Ŝ^2 can be substantial. 




We revisit each case from Section <ref>, beginning with a two-way ANOVA (Example <ref>). 


    Under ABCs (<ref>) and (<ref>) and (<ref>), the OLS SEs of all main effects under the cat-modified model  (<ref>) are less than or equal to those under the main-only model (<ref>): (α̂_0) ≤(α̂_0^M), (β̂_1,r) ≤(β̂_1,r^M) for all r=1,…,L_R, and (β̂_2,s) ≤(β̂_2,s^M) for all s=1,…,L_S.
    

Remarkably, Theorems <ref> and <ref> confirm that ABCs deliver the best possible result: adding cat-modifiers to the main-only model (<ref>) does not change the main effect estimates, but potentially decreases their SEs. Thus, analysts may include cat-modifiers “for free"—with no negative consequences for the main effects—while acquiring the ability to infer possibly heterogeneous, group-specific effects. The same occurs for categorical-continuous interactions, again with the equal-variance condition: 
 
     Under ABCs (<ref>), equal-variance (<ref>), and (<ref>), the OLS SE for the main x-effect under the cat-modified model (<ref>) is less than or equal to that under the main-only model (<ref>):  (α̂_1) ≤(α̂_1^M). 
    
    
    

This result applies in the context of Theorem <ref>, but analogous extensions are available for Theorems <ref> and <ref>; only the condition (<ref>) must be added.

These results make minimal assumptions about the true data-generating process and do not require independence or uncorrelatedness among the covariates. However, the OLS SEs are defined as usual, which implicitly refers to uncorrelated and homoskedastic error assumptions for both the main-only and cat-modified models. Thus, while Theorems <ref> and <ref> are direct statements about the SEs as statistics and do not require any assumptions on the error distributions, the utility of these results is clearly linked to these assumptions. 










 


 
 





 
 
















  










§ SIMULATIONS







 §.§ Validating invariance for estimation and inference

The first objective is to verify the theory of ABCs for estimation and inference invariance,  focusing on the conditions in Section <ref>. We consider both categorical-categorical interactions (Section <ref>) and categorical-continuous interactions (Section <ref>). 





  §.§.§ Categorical-categorical interactions

Given two categorical variables, say  and , what is the effect of including the  interaction term on the estimates and SEs for the  and  main effects? The theory of ABCs (Section <ref>) predicts that the estimates will be exactly the same, while the SEs may decrease if the interaction effect is sufficiently large. These results make no requirements on the data-generating process. Thus, we simulate data such that  1)  and  are dependent, 2) the errors are non-Gaussian, and 3)  ABCs are not satisfied. 

Let  and  be categorical variables with groups {, , , } and {, }, respectively; we use arbitrary labeling here to remain agnostic about particular race- or sex-specific effects in our synthetic data-generating process. For each of n=500 observations, we  draw each  assignment with ( π_a,  π_b,  π_c,  π_d) = (0.4, 0.3, 0.2, 0.1), and then draw the  assignment conditional on  with 
 ( π_uu,  π_vv)_·| r =  = (0.4, 0.6), 
 ( π_uu,  π_vv)_·| r =  = (0.6, 0.4), 
 ( π_uu,  π_vv)_·| r =  = (0.7, 0.3),  and 
 ( π_uu,  π_vv)_·| r =  = (0.2, 0.8). Thus,  and  are dependent, and marginally π_uu = π_vv = 0.5.  The response variable y is simulated with expectation (<ref>) with α_0 = 1, β_c = -1,  γ_b,vv = γ, and all other coefficients zero, or equivalently, 
μ(r,s) = 1   - 𝕀{r = } +  γ𝕀{r = , s = } 
plus t_4(0,1)-distributed errors. 
Crucially, γ controls the  magnitude of the   effect: we consider γ = 0 (no interaction effect), γ = 0.5 (moderate interaction effect; see the supplementary material),  and γ = 1.5 (large interaction effect).  
We repeat this process to create 500 synthetic datasets. 


For each simulated dataset, we fit the main-only model (<ref>) 
and the cat-modified model (<ref>) 
and compare the estimates and SEs for each main effect between the two models. These models are fit using ABCs, RGE (references  r =, s =), and STZ. This setting is favorable for RGE: the data-generating process satisfies RGE (β_a = 0, β_uu = 0, γ_a,uu = 0), but not ABCs, and the reference groups are the most abundant groups for both  and . To aid comparisons, we omit the main effects from the RGE reference groups, resulting in four main effects (β_b, β_c, β_d, β_vv) to compare between the main-only and cat-modified models for ABCs, RGE, and STZ. 
 

The estimates are in Figure <ref>. Under ABCs, all  and  main effects are exactly identical between the models that do and do not include the  interaction, confirming Theorem <ref>. This result persists regardless of the true data-generating process, including the magnitude of the interaction. No such invariance occurs for RGE or STZ: the inclusion of the interaction completely changes the estimates (and the interpretations) of the main effects. 




The SEs are in 
Figure <ref>. 
Under ABCs,  the addition of the  interaction  has virtually no impact on the main effect SEs. For larger γ, the main effect SEs are slightly smaller (about a 5% reduction) for cat-modified model, as expected. More substantial SE reductions occur for larger interactions (γ≥ 5), although such large interaction effects are not usually expected in practice. 
Again, no such results occur for RGE: the SEs are much larger for the model that includes  the  interaction, regardless of γ. 



These results must be interpreted carefully: the “main effects" under ABCs, RGE, or STZ target different functionals of μ(r,s). In fact, the OLS fitted values for μ̂(r,s) are identical under each identification (this is not the case for regularized regression). However, each identification puts forth “main effects" in both the main-only and cat-modified model. We argue that the main effects under ABCs are superior: the estimates are exactly invariant to the inclusion of () interactions and the SEs may decrease slightly. Uniquely, ABCs circumvent the traditional roadblocks to including interactions: the interpretations remain simple (and equitable) and there is no loss of statistical power for the main effects. 







  §.§.§ Categorical-continuous interactions

We now revise this analysis for categorical-continuous interactions: given categorical  and continuous , what is the effect of including the  interaction on the main x-effect? The theory of ABCs (Section <ref>) predicts that invariance for estimation and inference is contingent on  the equal-variance condition (<ref>). We investigate the sensitivity to this condition as well as to the magnitude of the interaction effect. 

To incorporate dependencies between  and , we simulate  as in Section <ref> and then simulate  conditional on : 
[| = ] ∼ 5  + σ_ac N(0,1), 
[| = ] ∼√(12) (0,1), 
[| = ] ∼ -5 + σ_act_8(0,1), and 
[| = ] ∼(1,1). 


    [| = r] ∼
        5  + σ_ac N(0,1)      r = 
    √(12) (0,1)     r = 
    
         -5 + σ_act_8(0,1)     r = 
    (1,1)     r =


Each  group features a unique distribution with varying means, so  and  are strongly dependent and highly correlated. 
Here, σ_ac controls the degree to which the equal-variance condition (<ref>) is violated: σ_ac = 1 is a mild violation (the race-specific population variances are identical, but the sample quantities σ̂_x[r]^2 are not) while σ_ac = 1.5 is a strong violation. The response variable y is simulated with expectation (<ref>) with α_0 = α_1 = 1, β_c = -1, and γ_b = γ, and all other coefficients zero, or equivalently, 
μ(x, r) = 1 + x  - 𝕀{r = } + γ x 𝕀{r = } 
plus t_4(0,1)-distributed errors. 
This data-generating process satisfies RGE (β_a = 0), but not ABCs, and includes non-Gaussian errors. Again, γ∈{0,0.5,1.5} determines the magnitude of the interaction effect.  We repeat this process to create 500 synthetic datasets. 


For each simulated dataset, we fit the main-only model (<ref>)  
and the cat-modified model (<ref>) 
and compare the estimates and SEs for main x-effect α_1 between the two models under ABCs, RGE, and STZ. 
The estimates are in Figure <ref>. Even with mild deviations from the equal-variance condition (<ref>), the x-effect estimates under ABCs are nearly identical between models that do and do not include the  interaction. Crucially, this invariance persists regardless of the true interaction effect magnitude γ. Under strong violations of (<ref>) and a strong interaction effect, Theorem <ref> no longer applies. However, as argued in Section <ref>, this behavior is appropriate: when (<ref>)  is strongly violated,  a one-unit change in x is not comparable for different  groups, so only the model that includes race-specific x-effects (via the  interaction) is appropriate. Finally, we note the absence of invariance for estimation with RGE or STZ. These estimators change dramatically when γ is moderate to large. Even when γ = 0—when classical consistency results for OLS should provide asymptotic invariance in this case—they do not match the invariance of ABCs.





The SEs are in Figure <ref>. As long as the violations of (<ref>) are mild (σ_ac = 1), the SEs of the x-effect, under ABCs, are 1) nearly identical between the main-only and cat-modified models when the true interaction effect is small  and 2) smaller for the cat-modified model  when the true interaction effect is large. Critically, including the  interaction under ABCs does not sacrifice any statistical power for the main x-effect, and in some cases enhances it. This is decisively not the case for RGE or STZ: when the true interaction effect is zero, adding the  interaction decreases statistical power for the main x-effect. 

The same caveats apply as in Section <ref>: RGE, STZ, and ABCs are targeting different functionals of   μ(x, r), but again we argue that the estimation and inference properties of the “main effects" are most ideal under ABCs. 







 
 
 




 









 §.§ Evaluating estimation and inference with cat-modifiers

We evaluate the practical impacts of the estimation invariance and enhanced power of ABCs. The goal is quantify the extent to which ABCs 1) maintain accurate estimates and precise uncertainty quantification when extraneous cat-modifiers are included and 2) improve estimation and reduce uncertainty when necessary cat-modifiers are included. The simulation design has three main features, described below.


First, we generate multiple, dependent categorical and continuous covariates. Dependent categorical variables  and  are generated as in  Section <ref>, while p=10 dependent continuous variables 
are generated as follows: x_j is drawn as in Section <ref> with σ_ac=1 for j=1,3,5,7,9 and  N(0,1) for j=2,4,6,8,10. Some x-variables are correlated with , which induces correlations among those x-variables with each other and with , while others are uncorrelated.  

Second, the regression coefficients are constructed  to satisfy both RGE and ABCs. 
These include an intercept α_0=1,  active main x-effects α_j = 1 for j=1,…,5,  main effects β_b = 1 and β_c = β_d = -1, and cat-modifiers γ_b,j = γ and γ_c,j = γ_d,j = -γ for j=1,…,5; the remaining coefficients are all zero. RGE is enforced because all reference group coefficients are zero  (β_a  = 0, β_uu = 0, γ_a,uu = 0, and γ_a, j = 0 for all j=1,…,p), while ABCs are satisfied for the population proportions ( π_a,  π_b,  π_c,  π_d) = (0.4, 0.3, 0.2, 0.1). Thus, it is meaningful to compare coefficient estimates and inference between RGE and ABCs (STZ is not satisfied and thus excluded). ABCs actually use the sample proportions and are at a slight disadvantage. All  main and interaction effects are zero, since this is the only way to satisfy both RGE and ABCs for a variable with two groups. 



Third, a parameter γ controls the magnitude of the cat-modifier () effect. We consider γ = 0 for extraneous cat-modifiers and γ = 1.5 for necessary cat-modifiers.  

Using these covariates and coefficient values, the response variable y is simulated with expectation (<ref>)  plus Gaussian errors and a signal-to-noise ratio of one. We vary the sample size n ∈{200, 500, 1000}  and repeat this process to create 500 synthetic datasets. 

For each synthetic dataset, we fit the main-only  model  and the cat-modified model  that includes  interactions with all continuous covariates and , both under ABCs and RGE. The main-only model is favored when γ = 0, while the opposite is true for γ > 0. Either way, the true data-generating process is sparse, so both models include many extraneous parameters. The main-only model includes 15 identifiable parameters (9 true signals)  while the cat-modified model includes 48 identifiable parameters. When γ = 0, the cat-modified model estimates 33 extraneous (identified) parameters; even when γ > 0, only 15 of those cat-modifier effects are nonzero.  



Evaluation primarily focuses on the  main x-effects  (α_1,…,α_10), which isolates the impacts of including  extraneous (γ = 0) or necessary (γ > 0) cat-modifiers on estimation and inference for the main effects. For benchmarking, we also include evaluations for all (main and cat-modifier) coefficients. Note that for the main-only model, RGE and ABCs produce main x-effect estimates and SEs that are identical and nearly identical, respectively. 

Estimation accuracy is evaluated by root mean squared error (RMSE) for the regression coefficients (Figure <ref>). The cat-modified model with ABCs preserves estimation accuracy of the main x-effects, even when (all 33) cat-modifier effects are included unnecessarily (Figure <ref>, top left). When some cat-modifiers are necessary, the cat-modified model with ABCs delivers slightly more accurate main x-effect estimates than the main-only models (Figure <ref>, bottom left). Neither result holds for RGE. By comparison, estimation accuracy across all coefficients 
(Figure <ref>, right) overwhelmingly favors the correctly-specified model (main-only for γ=0, cat-modified for γ = 1.5), regardless of RGE or ABCs. This result is not surprising, but rather serves as a contrast to emphasize the extraordinary robustness of main effect estimation accuracy for cat-modified models—but only under ABCs.  





Inference is evaluated by mean interval widths and empirical coverage for 95% confidence intervals for the regression coefficients (Figure <ref>); narrow intervals are preferred, subject to nominal coverage.  For ABCs, the cat-modified model offers nearly the same statistical power for the main x-effects as does the main-only model, even when (all 33) cat-modifier effects are included extraneously (Figure <ref>, top left). Compare that to inference for all coefficients (Figure <ref>, top right): here, the inclusion of extraneous cat-modifiers inflates interval widths by more than 300%. Clearly, this inferential robustness against extraneous cat-modifiers is a special property for 1) main effects and 2) ABCs.  When some cat-modifiers are necessary, the cat-modified model with ABCs  improves statistical power for the main x-effects compared to the main-only models. Again, no such results hold for RGE, for which the cat-modified model consistently sacrifices statistical power. Finally, as expected, main-only models fail to provide coverage for active cat-modified parameters (Figure <ref>, bottom right). 





The supplementary material includes additional results for smaller (n=200) and larger (n=1000) sample sizes; predictive evaluations based on RMSEs for μ( x,  c); comparisons between ABCs and RGE for lasso and ridge regression, also including an “overparametrized" version that does not impose any constraints; and modifications for σ_ac=1.5 that strongly violate the equal-variance condition (<ref>), with similar results as for σ_ac=1. 




 












§ APPLICATION

We apply cat-modified regression to assess heterogeneity among factors linked to STEM educational outcomes. Our dataset[Data management, access, and analysis are governed by data use
agreements and an Institutional Review Board–approved research protocol at the
University of Illinois Chicago.] links three administrative datasets to provide individual-level data for n=27,638 children in North Carolina (NC):   NC Detailed Birth Records, NC Blood Lead  Surveillance, and NC Standardized Testing Data; details are provided elsewhere <cit.>. The STEM educational outcome variable y_i is the end-of-4th-grade standardized math score for student i, centered and scaled by year of test (2010, 2011, or 2012). These math scores are linked with a rich collection of  demographic, social, and environmental exposure variables. The continuous covariates are  racial (residential) isolation (), which is a measure of structural racism based on neighborhood information; blood lead level (), which measures lead exposure; 
birthweight percentile (); mother's age at time of child's birth (); and exposure to the air pollutant _2.5  during the year prior to the exam (). The continuous covariates are centered and scaled. The categorical covariates are mother's race (), child's sex (), mother's education level (), and an indicator of economically disadvantaged () determined by participation in the National Lunch Program; see Table <ref> for categorical levels and proportions. 

Our linear regression analysis spans from main-only models to a variety of cat-modified models, expanding significantly upon the simple models from  Tables <ref> and <ref>. First, we establish a main-only model that includes each of these covariates (, , , , , , , , and ) but no interactions. 

The main-only model features a variety of interesting demographic, socio-economic, maternal, and environmental exposure variables, with  16 regression parameters (12 identified). Next, the race-modified model adds an interaction between  and every other covariate. 

This expansion allows for heterogeneous effects of each variable by race, thus providing insights into the myriad impacts of race on each child's life course and educational outcomes,  with  52 regression parameters (30 identified). Finally, the cat-modified model adds all pairwise categorical-continuous and categorical-categorical interactions. This instance of (<ref>) allows the fullest (pairwise) extent of heterogeneous effects across the rich collection of demographic and socio-economic variables (, , , and ), with 103 regression parameters (55 identified). We fit each of these models under ABCs and RGE (references , , lowest  (), and not ). 

While each model offers potential for insight, a critical limitation of popular identification approaches, especially RGE, is that the estimates, inference, and interpretations of the main effects are highly sensitive to the choice of cat-modifiers. To see this, we present the main effect OLS estimates and 95% confidences intervals across these models in Figure <ref>. With RGE (right), the main effects shift and the intervals widen considerably—with increases from 160% to 230% in interval widths—upon adding race- (blue) and other (red) cat-modifiers. These main effects and accompanying interaction effects (not shown) are anchored at the reference groups and refer to different functionals of μ( x,  c) under each model—even though the statistical output for the “main effects" is typically presented identically, regardless of any cat-modifiers. Thus, while cat-modified models are essential for heterogeneous effects, there is a cost incurred under RGE: each additional  cat-modifier requires careful re-consideration of the main  and interaction effects, which impedes statistical analysis and undermines interpretability. 





The invariance of ABCs resolves these limitations: estimation and inference for the main effects (Figure <ref>, left) are nearly identical across these substantially different models. This occurs despite strong dependencies among the covariates (and interactions) with both continuous and categorical variables. ABCs effectively decouple the main effects from the cat-modifiers: even adding 87 parameters (43 identified) from the main-only model to obtain the cat-modified model does not lessen, and in some cases increases the statistical power for the main effects. With ABCs, the statistical analyst may consider these or other cat-modified models without compromising or complicating inferences for the main effects. 


The full regression output from the cat-modified model with ABCs is in Tables <ref> and <ref>. 
Lower math scores are strongly (p < 0.01) associated with racial (residential) isolation, lead exposure, lower (mother's) education levels, and occur for non-Hispanic Black and economically disadvantaged students; higher math scores are strongly associated with birthweight percentile, mother's age, and the opposing categories from above. ABCs provide output for all levels of all categorical variables, thus eliminating the presentation bias of RGE that presents all output relative to the reference groups (White, Male, etc.). For categorical variables with ABCs, the estimates and SEs are directly related to the abundances: for instance, categorical variables with equal proportions such as , the main and -continuous interaction estimates are equal and opposite with identical SEs for  Male and Female. 


The regression output strongly supports heterogeneous effects, most notably via mother's education level and with intersectionality of race and sex (e.g., ). 



    











Finally, we simplify the heavily-parametrized cat-modified model by fitting a lasso regression under ABCs; λ is selected using 10-fold cross-validation and the one-standard-error rule <cit.>. The selected main effects  (, , , , , , and ) match the conclusions from Figure <ref>. Among interactions,  coefficients from , , , , and  are selected. The accompanying coefficients of these cat-modifiers suggest that some positive effects are not as beneficial for minoritized groups: the positive effect of  mother's education () are attenuated for  Black and Hispanic students, while the benefits of mother's age are less so for lower mother's education, Black, or economically disadvantaged students. 











§ CONCLUSION


To encourage and enable statistical analysis of heterogeneous effects, we analyzed and advocated ABCs—an alternative parametrization and estimation strategy for cat-modified models that include categorical-continuous or categorical-categorical interactions. Unlike default methods, ABCs allow the inclusion of cat-modifiers “for free": there is virtually no impact on the main effect estimates, while main effect inference is stable or more powerful. We rigorously proved these estimation and inference invariance properties and validated them empirically with extensive simulation studies. We also provided strategies for estimation and inference, including both generalized and regularized regression. Finally, we applied these tools to analyze STEM educational outcomes and showed how ABCs facilitate  identification and estimation of (demographic) heterogeneous effects without incurring any costs—in estimation, inference, or interpretation—for the main effects.  



Despite these many advantages, we note several caveats. First, ABCs may increase susceptibility to p-hacking. Because ABCs facilitate the inclusion of interactions, and with a large enumeration of potential interactions,  there is a heightened potential for both discovery and false discovery. Proper statistical analyses require careful consideration of hypothesis tests with  multiple testing corrections
as appropriate.  Second, ABCs cannot guarantee that cat-modifiers will be (practically or statistically) significant. Detection of heterogeneous effects often requires well-designed studies or large sample sizes. 
Third, our invariance results apply for least squares estimation, but not more general loss functions. Finally, many categorical variables, especially race, sex, and other protected groups, are susceptible to misinterpetation, inaccurate labelings, and exclusions of small groups. 


























 


§ ACKNOWLEDGEMENTS

We thank Virginia Baskin,  Caleb Fikes, Prayag Gordy, and Jai Uparkar for helpful discussions and their contributions to software development.



chicago









figuresection
tablesection
equationsection





 Supplement to 

    “Facilitating heterogeneous effect estimation via statistically efficient categorical modifiers"
    
      0
 
 Daniel R. Kowal
    
    





    This supplementary file includes proofs of all results (Section <ref>), details for generalized linear models (Section <ref>), additional simulation results (Section <ref>), and additional details and analyses of the North Carolina education data (Section <ref>).





§ PROOFS
 

We first provide a sketch of the general proof technique. Our results require only basic linear algebra, but the notation can be cumbersome. Here, the goal is to provide clear intuition for our results and to put forth a blueprint to analyze similar invariance properties in other settings. 

Consider two generic but nested models:

     

    

The task is to establish conditions under which the OLS estimates of the coefficients on X_* are unchanged by the addition of X_1, with X_0 also present in both models.  In our typical setting,   X_* is a matrix of (continuous) covariates, X_0 is a matrix of categorical indicator variables, and  X_1  contains cat-modifiers. Crucially, for identifiable estimation and inference, these matrices involving categorical covariates or cat-modifiers must already be parametrized to enforce the identifiable constraints, such as omitting certain columns for RGE or applying the QR reparametrization from Section <ref> for ABCs. 

The most relevant classical result is due to Frisch and Waugh (1933) and Lovell (1963): 


 Frisch-Waugh-Lovell (FWL) Theorem:
    
    For a partition of the n × p covariate matrix X = ( X_0: X_1) into p_0 and p_1 columns, the partition of the ordinary least squares estimator β̂ = (β̂_0^⊤, β̂_1^⊤)^⊤ satisfies β̂_0 = ( X_0^⊤ E_01)^-1 E_01^⊤ y = ( E_01^⊤ E_01)^-1 E_01^⊤ y, where E_01 = ( I_n -  H_X_1)  X_0 is the n × p_0 matrix of residuals from regressing each column of X_0 on X_1, H_X_1 =  X_1 ( X_1^⊤ X_1)^-1 X_1^⊤ is the corresponding hat matrix for X_1, and y = (y_1,…,y_n)^⊤ is the vector of outcomes. 

 

Applying the FWL Theorem, our target result occurs when  = , for which a sufficient condition is X_*^⊤ E_10 = 0
where E_10 = . More formally, let H_0  X_0( X_0^⊤ X_0)^-1 X_0^⊤ be the hat matrix for the covariates X_0 that are always included. Then the sufficient condition is 

    X_*^⊤ ( X_1 -  H_0  X_1) =  0

or equivalently, X_*^⊤ X_1 - ( H_0 X_*)^⊤ X_1 =  0, if we prefer to consider regressing X_0 on X_* instead of X_1. 
 In the simpler case without a common X_0 term, the requirement simplifies to X_*^⊤ X_1 =  0, where the role orthogonality is now abundantly clear. 

In the presence of ABCs (or other linear constraints), we apply the reparametrization from Section <ref> that replaces X_1 with X_1  Q_π̂ to enforce the constraints. The main condition (<ref>) is now 

    X_*^⊤ ( X_1 -  H_0  X_1) Q_π̂ =  0.

The key observation is that   A_π̂ Q_π̂ =  0 by construction; this is true for the QR-based approach with any constraints of the form A_π̂θ =  0, including but not limited to ABCs. Thus, the general  requirement is to show that X_*^⊤ ( X_1 -  H_0  X_1) is  row-wise proportional to A_π̂, which produces the necessary zeros. 

We apply this strategy for Theorems <ref>–<ref>, but prove the main results in sequence.





    For simplicity, we prove this result for the case of (<ref>), but the same ideas apply more generally. 
    
    It is sufficient to show that  𝔼_π̂(β_1,R + β_2,S + γ_RS) = 0. Direct application of (<ref>) implies 𝔼_π̂(β_1,R + β_2,S + γ_RS) = 𝔼_π̂_R(β_1,R) + 𝔼_π̂_S(β_2,S) + 𝔼_π̂(γ_RS) = 𝔼_π̂(γ_RS), and further simplifying, 𝔼_π̂(γ_RS) = ∑_r=1^L_R∑_s=1^L_Sπ̂_rsγ_rs = 0 since the internal summation is zero for all r by (<ref>).



    We prove this result for the case of (<ref>) for simplicity. Applying (<ref>) to all but r=1, we have ∑_r=1^L_Rπ̂_rsγ_rs = 0 for s=1,…,L_S and thus γ_1s = -π̂_1s^-1∑_r=2^L_Rπ̂_rsγ_rs. The conditional expectation is then 
    𝔼_π̂_S | R=1(γ_RS) = ∑_s=1^L_Sπ̂_1sγ_1s
        = -∑_s=1^L_S
        ∑_r=2^L_Rπ̂_rsγ_rs
        =∑_r=2^L_R∑_s=1^L_Sπ̂_rsγ_rs
        =0 
    since the internal summation equals zero for all r>1. 




    Under OLS, y̅ equals the sample mean of the fitted values {ŷ_i}_i=1^n; this is true for ABCs, RGE, STZ, etc. Then we simplify: 
    
    y̅   = n^-1∑_i=1^n ŷ_i = n^-1∑_i=1^n(α̂_0 +  x_i^⊤α̂ + ∑_k=1^K β̂_k, c_k + ∑_k=1^K-1∑_k'=k+1^K γ̂_k,k', c_k, c_k' )
       = α̂_0 + x̅^⊤α̂ +  ∑_k=1^K ∑_c_k =1^L_kπ̂_k, c_kβ̂_k, c_k + ∑_k=1^K-1∑_k'=k+1^K ∑_c_k =1^L_k∑_c_k' =1^L_k'π̂_k,k', c_k, c_k'γ̂_k,k', c_k, c_k'
       = α̂_0

    since the continuous covariates are centered (x̅ =  0) and the main categorical effects and categorical-categorical interactions satisfy ABCs, so the interior summations equal zero for all k, k'. 






    Following the  and  terminology from Example <ref>,  define the design matrix by letting 1 be an n-dimensional vector of ones, Z_1 the n × L_R matrix of   indicators with  entries [ Z_1]_ir = 1 if r_i = r and zero otherwise, and Z_2 n × L_S matrix of   indicators  with entries [ Z_2]_is = 1 if s_i = s and zero otherwise. Similarly, let Z_12 be the n × L_RL_S matrix with indicators for the interaction terms. Consider the cross-produces of each main effect with the interaction matrix. 
    First, 1^⊤ Z_12 is the 1 × L_RL_S matrix where each entry is the joint total by  and , i.e., ∑_i=1^n 𝕀(r_i = r, s_i = s) for each r,s combination. 
    Next, Z_1^⊤ Z_12 is L_R × L_RL_S, where each row r includes the totals  ∑_i=1^n 𝕀(r_i = r, s_i = s) for all s=1,…,L_S but zeros for columns with other  groups, r'  r. Similarly, Z_2^⊤ Z_12 is L_S × L_RL_S, where each row s includes the totals  ∑_i=1^n 𝕀(r_i = r, s_i = s) for all r=1,…,L_R but  zeros for columns with other  groups, s'  s. 

    Estimation invariance occurs when these cross-products are zero. However, we must also account for identifiability constraints. Following Section <ref>,   Z_12 is replaced by Z_12 Q_π̂, where A_π̂ Q_π̂ =  0 and  A_π̂ encodes the constraints on the interaction coefficients. Thus, it suffices to show that 1^⊤ Z_12 Q_π̂ =  0, Z_1^⊤ Z_12 Q_π̂ =  0, and Z_2^⊤ Z_12 Q_π̂ =  0, with each zero of the appropriate dimension. 
    For ABCs, the latter two cross-products, when scaled by n^-1, exactly match the joint ABCs 
    (<ref>) in the form of A_π̂, and thus are zero upon post-multiplication by Q_π̂. Similarly, the first cross-product is also zero by applying the arguments from Lemma <ref>.
    








    Let  y = (y_1,…,y_n)^⊤, 
    x = (x_1,…, x_n)^⊤, and Z be the matrix of categorical () indicators with entries [ Z]_ir = 1 if r_i = r and zero otherwise. The cat-modifier term is Z_X =  D_X  Z and D_X = ( x). The goal is to show that, under the stated conditions, (<ref>) holds with x =  X_*, X_1 =  Z_X, and H_0  =  H_Z =  Z( Z^⊤ Z)^-1 Z^⊤ is the hat matrix for the categorical covariate. 
    
    For clarity, we provide more detailed results en route. Applying the FWL Theorem, the estimated coefficients under (<ref>) satisfy 
    α̂_1^M = ( x^⊤ê_x∼ r)^-1ê_x∼ r^⊤ y, 
    where  ê_x∼ r is the vector of residuals from regressing the continuous variable {x}_i=1^n on the categorical variable {r_i}_i=1^n (i.e., Z). Similarly, the estimated coefficients under (<ref>) satisfy 
    α̂_1 = ( x^⊤ê_x∼ r + Z_XQ)^-1ê_x∼ r + Z_XQ^⊤ y, 
     where ê_x∼ r + Z_XQ are the residuals from regressing the continuous variable {x}_i=1^n on the categorical variable {r_i}_i=1^n (i.e., Z) and the reparametrized interaction term that enforces  ABCs,  Z_XQ =  Z_X  Q_-(1:m) (see Section <ref>). Thus, it suffices to show that ê_x∼ r = ê_x∼ r + Z_XQ, which occurs when the additional (interaction) coefficients from the latter model, say b̂_Z_XQ (corresponding to Z_XQ), are identically zero. Again using  the FWL Theorem, these estimated coefficients are b̂_Z_XQ =   Q_-(1:m) ( Z_XQ^⊤ E_Z_XQ)^-1 E_Z_XQ^⊤ x, where E_Z_XQ is the matrix of residuals from regressing Z_XQ on Z, i.e., E_Z_XQ =  Z_XQ -  H_Z  Z_XQ. Thus, showing x^⊤ E_Z_XQ =  0 is sufficient, and factoring x^⊤ E_Z_XQ = ( x^⊤ Z_X -  x^⊤ H_Z  Z_X)  Q_-(1:m) shows the connection with (<ref>).
    
     First, observe that  x^⊤ Z_X =  x^⊤ D_X  Z = (s_x[1]^2,…, 
     s_x[L_R]^2) is the vector of s_x[r]^2 = ∑_r_i=r x_i^2 across groups. Next, observe that ( Z^⊤ Z)^-1 Z^⊤ Z_X = ({x̅_r}_r) contains the sample means of {x}_i=1^n by each group r, and therefore x^⊤ H_Z  Z_X =  x^⊤ Z ({x̅_r}_r) = (n_1 x̅_1^2, …, n_L_Rx̅_L_R^2) with n_r = n π̂_r. Combining these results, we have x^⊤ E_Z_XQ =  v^⊤ Q_-(1:m), where v^⊤ = (s_1^2 - n_1 x̅_1^2,…, s_L_R^2 - n_L_Rx̅_L_R^2) = n ( π̂_1 σ̂_x[1]^2, …,π̂_L_Rσ̂_x[L_R]^2) = n σ̂_x[1]^2 π̂^⊤ under the assumption that σ̂_x[r]^2 = σ̂_x[1]^2 is common for all r, which is precisely the equal-variance condition (<ref>). Finally, the definition of Q_-(1:m) via  ABCs implies that π̂^⊤ Q_-(1:m) =  0, which proves the result.





Let X denote the n × p matrix of continuous covariates, Z  the matrix of categorical dummy variables with entries [ Z]_ir = 1 if r_i = r and zero otherwise, and Z_XQ = ( Z_X_1Q,…,  Z_X_pQ) with Z_X_jQ =  Z_X_j Q_-(1:m), 
 Z_X_j =  D_X_j Z, and D_X_j = ( x_j). By the FWL Theorem, it suffices to show that E_M =  E, where E_M = ( I_n -  H_Z) X are the residuals from regressing each column of X on Z and E are similarly the residuals from regressing each column of X on Z and Z_XQ. Thus, it is sufficient to show that the coefficients associated with Z_XQ in the latter regression are identically zero. Again using the FWL Theorem, we see that this occurs whenever X^⊤ E_Z_XQ =  0, where E_Z_XQ =  Z_XQ -  H_Z  Z_XQ = ( Z_X_1Q -  H_Z  Z_X_1Q, …,  Z_X_pQ -  H_Z  Z_X_pQ). Noticing that X^⊤ E_Z_XQ = ( X^⊤ ( Z_X_1Q -  H_Z  Z_X_1Q), …,  X^⊤( Z_X_pQ -  H_Z  Z_X_pQ)), we consider the individual components x_h^⊤( Z_X_jQ -  H_Z  Z_X_jQ) = ( x_h^⊤ D_x_j Z -  x_h^⊤ H_Z  D_x_j Z)  Q_-(1:m), each of which must equal the zero vector with dimension equal to the number of categories. Noting that x_h^⊤ D_x_j Z = (…, s_r(j,h),…) with s_r(j,h) = ∑_r_i=r x_ijx_ih and x_h^⊤ H_Z  D_x_j Z = (…, n_r x̅_r(j) x̅_r(h), …) with x̅_r(j) = n_r^-1∑_r_i=r x_ij, we apply the same arguments as in Theorem <ref>. 





    Let X_0 be the matrix of covariates that includes X_-1 (i.e., all covariates but x_1) and the categorical (race) indicators Z with entries [ Z]_ir = 1 if r_i = r and zero otherwise, and let H_0  X_0( X_0^⊤ X_0)^-1 X_0^⊤ be its hat matrix. For the interaction terms, let Z_x_1Q =  Z_x_1 Q_π̂ where Z_x_1 =  D_x_1 Z, D_x_1 = ( x_1), and π̂^⊤ Q_π̂ =  0 enforces the ABCs for the interaction terms (see Section <ref>). Now, it is sufficient to show that ( x_1^⊤ Z_x_1 -  x_1^⊤ H_0  Z_x_1) Q_π̂ =  0 as in (<ref>). First, observe that x_1^⊤ Z_x_1 = (⋯ s_x_1[r]^2 ⋯ ). Second, x_1^⊤ H_0  Z_x_1 = x̂_1^⊤ D_x_1 Z = (⋯∑_r_i = rx̂_i1 x_i1⋯ ) where x̂_1 =   H_0  x. Combining these results, we see that v^⊤ x_1^⊤ Z_x_1 -  x_1^⊤ H_0  Z_x_1 = (⋯∑_r_i = r (x_i1^2 - x_i1x̂_i1) ⋯). Consider the interior terms for each r:  ∑_r_i = r (x_i1^2 - x_i1x̂_i1) =  ∑_r_i = r x_i1ê_i1 = n_r_r(ê_1, x_1), where the latter equality holds because ∑_r_i=rê_i1 = 0 for each r due to the inclusion of Z. Thus, v^⊤ = (⋯ n_r_r(ê_1, x_1) ⋯ ) = k(⋯ n_r ⋯) for some constant k that does not depend on r, which implies that v^⊤ Q_π̂ = nk π̂^⊤ Q_π̂ =  0 under ABCs.




    The proof of Theorem <ref> establishes orthogonality of ABCs-constrained interaction to the main effects under the same conditions. Given this orthogonality, the remainder of the proof follows the proof of Theorem <ref> and is omitted for brevity. 




    
    Applying the same arguments from the proof of Theorem <ref>, the variances satisfy 
        (α̂_1^M) = σ_M^2 ( x^⊤ê_x∼ r)^-1 and 
        (α̂_1) = σ^2 ( x^⊤ê_x∼ r + Z_XQ)^-1, 
    where σ_M^2 is the error variance from the main-only model and σ^2 is the error variance from the cat-modified model, assuming uncorrelated and homoskedastic errors in both models. These error assumptions are not required to prove the result, but do motivate the definition of the SE. Under the equal-variance condition (<ref>), we previously showed  that ê_x∼ r = ê_x∼ r + Z_XQ. Thus, the only difference in the variances of the estimators occurs because of the error variances, i.e., 
    (α̂_1)/(α̂_1^M)  =  σ^2/σ_M^2. The SEs  substitute point estimates for σ_M and σ: (α̂_1^M) = Ŝ_M √(( x^⊤ê_x∼ r)^-1) and similarly,
        
    (α̂_1)    = Ŝ√(( x^⊤ê_x∼ r + Z_XQ)^-1)
       = Ŝ√(( x^⊤ê_x∼ r)^-1)
       = Ŝ/Ŝ_M(α̂_1^M) 
       ≤(α̂_1^M)

    since Ŝ≤Ŝ_M under (<ref>). 







§ GENERALIZED LINEAR MODELS WITH ABCS
 

Generalized linear models (GLMs) are immensely useful for regression analysis with a variety of data types, including continuous, count, binary, and categorical data. Broadly, GLMs require a choice of data distribution (e.g., Gaussian, Poisson, Bernoulli, etc.) and a link function g that replaces the expectation of Y, say μ( x,  c)  with a transformed version, say g{μ( x,  c)}, in the cat-modified model (<ref>) (similarly for the main-only model (<ref>)). With categorical covariates and cat-modifiers, identification constraints are needed for the regression coefficients exactly as in the ordinary (untransformed) linear model. ABCs again provide a suitable identification strategy, with straightforward estimation and inference: the loss function ℒ in Section <ref>  is specified to incorporate the appropriate negative log-likelihood and link function. 

More subtly, the presence of the link function g implies that interpretations of the coefficients will be different from those in the  ordinary linear model (Section <ref>). For the main x_j-effects,  recall that α_j = 𝔼_π̂ (α_j + ∑_k=1^K γ_j, k, C_k) under ABCs (<ref>), regardless of the data distribution or the link function. When the link g is not the identity,  it is no longer the case that the internal quantity in the expectation equals μ_x_j'( C), and thus the previous interpretation from (<ref>) requires modifications. Most generally, cat-modified GLMs satisfy

    g{μ(x_j+1,  x_-j,  c)} - g{μ(x_j,  x_-j,  c)} = α_j + ∑_k=1^K γ_j, k, c_k,

so under ABCs (<ref>) the main x_j-effect is 

    α_j = 𝔼_π̂ [
        g{μ(x_j+1,  x_-j,  C)} - g{μ(x_j,  x_-j,  C)}].

As with ordinary linear regression, ABCs identify each main effect as a group-averaged comparison between expectations at (x_j+1,  x_-j) and (x_j,  x_-j). The main differences for GLMs is the presence of the link function g within this comparison. 


For clarity, we provide interpretations for logistic and Poisson regression. For binary data Y ∈{0,1}, μ( x,  c) equals the probability that Y=1 and logistic regression specifies g as the logit link, g(t) = log{t/(1-t)}. Now, (<ref>) simplifies to the log-odds-ratio: 

    g{μ(x_j+1,  x_-j,  c)} - g{μ(x_j,  x_-j,  c)} = log[{μ(x_j+1,  x_-j,  c)}/{μ(x_j,  x_-j,  c)}]

where {μ( x,  c)} = μ( x,  c)/{1 - μ( x,  c)}. Thus, α_j is the group-averaged log-odds-ratio for x_j. This interpretation is natural: for the main-only logistic regression model, α_j is simply the log-odds-ratio for x_j. 
Similarly, for Poisson regression with Y∈{0,1,…,}, μ( x,  c) is the expectation of Y and g(t) = log(t), so (<ref>) is a log-ratio:

    g{μ(x_j+1,  x_-j,  c)} - g{μ(x_j,  x_-j,  c)} = log{μ(x_j+1,  x_-j,  c)/μ(x_j,  x_-j,  c)}.

Here, α_j is the group-averaged log-ratio. Finally, we note that both of these terms involve group-averaged quantities on the log-scale. Thus, it may be more natural to consider exponentiated versions on the μ-scale, so the group-averages become weighted geometric means. 



In both cases, a logarithmic term is linked to α_j + ∑_k=1^K γ_j, k, c_k via (<ref>). Considering the exponentiated version, along with an averaging operation, we notice that 

    exp(α_j)    = exp{𝔼_π̂ (α_j + ∑_k=1^K γ_j, k, C_k)} = exp{∑_k=1^K ∑_ℓ=1^L_kπ̂_k,ℓK^-1(α_j + Kγ_j, k, C_k)}
       = ∏_k=1^K ∏_ℓ=1^L_kexp(α_j + Kγ_j, k, C_k)^π̂_k,ℓK^-1

which 
is a weighted geometric mean of α_j + Kγ_j, k, C_k. 







 










  §.§.§ Logistic regression

Logistic regression models a binary outcome y∈{0,1} as a Bernoulli random variable with “success" probabilities determined by the log-odds of  the linear term μ:

    {ℙ(y = 1 |   X = x, κ = k)} = log{ℙ(y = 1 |   X = x, κ = k)/ℙ(y = 0 |   X = x, κ = k)}   = μ(x, k)

Clearly, the odds are expressed by (y = 1 |   X = x, κ = k) = exp{μ(x,k)}, so that the odds ratio is linked to the “slopes":

    (y = 1 |   X = x + 1, κ = k)/(y = 1 |   X = x, κ = k)  = exp{μ_x'(k)} = exp(α_1 + β_k^*).

 As a consequence, the same issues arise for logistic regression as in linear regression: selecting a baseline group for the race variable k requires that all estimation and interpretation are with respect to the baseline group, which for race is typically NHW.  

Instead, suppose the ABCs are enforced. The previous interpretations require modifications, since the linear term appears on the log-odds scale. First, consider the common slope term α_1. Since α_1 = 𝔼_π̂{μ_x'(κ)} under the ABCs, we can derive  from (<ref>) that

    exp(α_1) = exp[ 𝔼_π̂{μ_x'(κ)}]  = ∏_k=1^K exp{μ_x'(k)}^π̂_k = ∏_k=1^K exp(α_1 + β_k^*)^π̂_k

which, since ∑_k=1^K  π̂_k = 1, is equal to the weighted geometric mean  of the race-specific odds-ratios for covariate x with weights given by the category proportions {π̂_k}_k=1^K. 

Naturally, the previous interpretations progress as before, but now in the context of (i) odds-ratios and (ii) weighted geometric means. 


 




§ ADDITIONAL SIMULATION RESULTS




 




















§ ADDITIONAL APPLICATION DETAILS












 













 








 






 Paper Title: Efficient subgroup testing in change-plane models.


#1 1
[figure]name=Fig.,labelsep=period



0
	
	
	Efficient subgroup testing in change-plane models
    
		Xu Liu 

		School of Statistics and Management,

		Shanghai University of Finance and Economics, Shanghai 200433, China

		Jian Huang 

		Department of Applied Mathematics,

		The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China

		Yong Zhou 

		Academy of Statistics and Interdisciplinary Sciences,

		East China Normal University, Shanghai 200062, China

		Feipeng Zhang

		School of Economics and Finance, 

		Xi'an Jiaotong University, Xi'an 710049, China

		Panpan RenCorresponding author. Email address: panpanren@stu.sufe.edu.cn 

		School of Statistics and Management,

		Shanghai University of Finance and Economics, Shanghai 200433, China

	
    
=======================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================

 

1
	

		Efficient subgroup testing in change-plane models
	

	 


§ ABSTRACT 
	Considered here is a hypothesis test for the coefficients in the change-plane regression models to detect the existence of a change plane. 
  The test that is considered is from the class of test problems in which some parameters are not identifiable under the null hypothesis. The classic exponential average tests do not work well in practice. To overcome this drawback, a novel test statistic is proposed by taking the weighted average of the squared score test statistic (WAST) over the grouping parameter's space, which has a closed form from the perspective of the conjugate priors when an appropriate weight is chosen. The WAST significantly improves the power in practice, particularly in cases where the number of the grouping variables is large. The asymptotic distributions of the WAST are derived under the null and alternative hypotheses. The approximation of critical value by the bootstrap method is investigated, which is theoretically guaranteed. Furthermore, the proposed test method is naturally extended to the generalized estimating equation (GEE) framework, as well as multiple change planes that can test if there are three or more subgroups. The WAST performs well in simulation studies, and its performance is further validated by applying it to real datasets.


Keywords:  Nonstandard tests; Quantile regression; Subgroup detection; U-statistics


1.45 


§ INTRODUCTION



Subgroup analysis has emerged as a powerful tool in diverse fields covering public management, clinical medicine and the physical and social sciences. 
In public management, the effect of a certain policy may be heterogeneous, which means that the same policy can have distinct impacts on different subpopulations. 
Ignoring a population's heterogeneity in classical models may cause biased estimates, and identifying subgroups of individuals is the core problem. 
Procedures for estimating and identifying subgroups have been extensively studied based on various models <cit.>. Before identifying subgroups, an appropriate statistic is required to test for their existence. 
However, this test problem has received relatively little attention, especially when there are large number of grouping variables 
that can divide the population into subgroups with different associations between predictors and responses. 
The existing literature for the testing problem is outlined below, focusing on several classical change-plane regression models. Therefore, herein, we delve into subgroup testing in the change-plane regression models, which belongs to the class of hypothesis tests when some parameters are not identifiable under the null hypothesis.

To motivate our work, we look at the data from the 2013 China Health and Retirement Longitudinal Study (CHARLS). It is widely acknowledged that population aging is a significant global social issue, and China has been facing this challenge since 2001.  In response, the Chinese government has established a social old-age insurance system to alleviate the burden on adult children. 
As is known, child support and social pension are two major sources of income for rural elderly in China. 
Does the pension income from the New Rural Social Pension Insurance (NRSPI) effectively alleviate the financial burden on adult children of the elderly? 
Many scholars have offered different answers to this question <cit.>. 
Addressing this question can help find out the shortcomings of the current pension insurance system and make improvements. 
Therefore, it is crucial to identify whether there is a subpopulation in which pension income has a distinct impact on support income of the elderly. 
This paper considers a dataset from CHARLS in Shanxi Province. 
The support income of the elderly in Shanxi Province, which is considered as the response variable, displays heavy-tailed and asymmetry characteristics (see Figure D.1 in the Supplementary Material), prompting us to develop a quantile change-plane regression model. 

For the different types of data and goals, there are many change-plane regressions considered in the literature.
Here, we give some common and useful examples.
Quantile regression. 
	<cit.> and <cit.> considered single-index thresholding in quantile change-plane regression:
	
    Q_Y_i(τ|_i, _i, _i) = _i(τ)+_i(τ)(_i(τ)≥0),

	where Q_Y(τ|·) denotes the τth conditional quantile of the response Y given covariates , , and , τ∈(0, 1) is the quantile level of interest, _i is a subset of _i, and (·) denotes the indicator function.
	For fixed , the score function with respect to  at = is
	
    ψ(_i, , , )=
    		(_i(τ)≥0)[(Y_i-_i(τ)≤ 0)-τ]_i,

	where =(τ) is the estimate of =(τ) when = according to
	
    ∑_i=1^nψ_1(_i, ) = ∑_i=1^n[(Y_i-_i(τ) ≤ 0)-τ]_i=.

	
	


 Logistic regression model. <cit.> studied the logistic change-plane regression model, which belongs to generalized linear models (GLMs) with the likelihood
	
    exp[ Y_iμ_i - c(μ_i)]a(Y_i),   μ_i =   _i + _i(_i≥0),

	where μ_i is the canonical parameter, and a(·) and c(·) are known functions. 
For fixed , the score function with respect to  at = is
	
    ψ(_i, , , )= (Z_i≥θ)[Y_i-∂ c(_i)/∂μ_i]_i,

	where  is the estimate of  when = according to
	
    ∑_i=1^nψ_1(_i, ) = ∑_i=1^n[Y_i-∂ c(_i)/∂μ_i]_i=.





Probit regression model.
	<cit.> considered the probit change-plane regression model with likelihood
	
    Φ(_i+_i(Z_i≥θ))^Y_iΦ(-_i-_i(Z_i≥θ))^1-Y_i,

	where Φ(·) is the cumulative distribution function of the standard normal distribution. 
For fixed , the score function with respect to  at = is
	
    ψ(_i, , , )=
    		(Z_i≥θ)[Y_iϕ(_i)/Φ(_i) - (1-Y_i)ϕ(-_i)/Φ(-_i)]_i,

	where ϕ(·) is the density of the standard normal distribution, and  is the estimate of  when = according to
	
    ∑_i=1^nψ_1(_i, ) = ∑_i=1^n[Y_iϕ(_i)/Φ(_i) - (1-Y_i)ϕ(-_i)/Φ(-_i)]_i=.

	



 Semiparametric model.
	<cit.> considered the semiparametric change-plane model
	
    Y_i = γ(_i) +β X_i (_i≥0)+_i,

	where γ() is an unknown baseline mean function for patients in treatment X=0, X_i∈{0, 1}, and (_i|X_i, _i)=0. 
For fixed , the score function with respect to  at = is
	
    ψ(_i, , , )=
    		(_i≥0)[X_i-π(_i, _1)][Y_i-γ(_i, _2)],

	where =(_1, _2) is the estimate of =(_1, _2) when = according to
	
    ∑_i=1^nψ_1(_i, )
    		=∑_i=1^n(
    		[ D__1(_i){X_i-π(_i, _1)}; D__2(_i){Y_i-γ(_i, _2)} ])=

	with D__1(_i)=[π(_i, _1)(1-π(_i, _1))]^-1∂π(_i, _1)/∂_1 and D__2(_i)=∂γ(_i, _2)/∂_2. Following <cit.>, we emphasize that either the baseline mean function γ(, _2) or the propensity model π(_i, _1) is specified correctly, but not necessarily both.


The goal, subgroup or change-plane test, considered herein is to test whether  is  or not, i.e.,

    H_0: =  versus   H_1: ≠,

where the nuisance parameter  is only identifiable under the alternative hypothesis.
For convenience of expression, we call  the ,  the ,  the grouping difference variable, and  the . The grouping difference parameter measures the effect difference between two subgroups. _i is the baseline model,  is the baseline variable, and  is called the effect of the baseline group.

From examples 1–4, it is natural to extend these classical models to the generalized estimating equation (GEE) framework.
Let the observed data consist of {_i=(Y_i,_i,_i,_i), i=1, ⋯, n}, which are n independent and identically distributed copies of =(Y,,,).
Consider the GEEs [(≥ 0)ψ_0(, , ,)]= and ψ_1(, )= and the corresponding empirical estimating functions

    Ψ_n(, , )=   ∑_i=1^n(_i≥ 0)ψ_0(_i, , ,), 
    Ψ_1n()=   ∑_i=1^nψ_1(_i, ),

where =(α_1, ⋯, α_r)∈Θ_α⊆ℝ^r, =(β_1, ⋯, β_p)∈Θ_β⊆ℝ^p, and =(θ_1, ⋯, θ_q)∈Θ_θ⊆ℝ^q are unknown parameters. Here, ψ_0(, , ,)
is independent of both  and . 
For ease of expression, we denote ψ(, , , )=(≥ 0)ψ_0(, , ,) and ψ_0(, )=ψ_0(, , ,).
In many hypothesis testing problems,  is a nuisance parameter, such as in the examples above. The functions (<ref>) for considering testing problem (<ref>) are flexible, with many existing models included as special cases, such as examples 1–4.

In the ordinary testing problem in which there is no identifiability problem under either the null or alternative hypothesis, the classic Wald- and score-type tests perform well. When nuisance parameters are present, <cit.> and <cit.> studied the optimal tests based on the weighted average power criterion, which was originally introduced by <cit.> when studying the likelihood ratio test under regularity conditions. <cit.> investigated the supremum of the squared score test statistic for mixture models, which is a special case of the optimal tests. <cit.>, and <cit.> considered nonstandard tests for semiparametric models with the application of censoring data. <cit.> studied a score-type test to detect the existence of social network dependence with time-to-event data.


All the aforementioned methods require taking the supremum of a classic Wald- or score-type test statistic. 


The large feasible region of calculating the supremum may cause these tests to not perform well, and they are time-consuming computationally.
To address these limitations, this paper introduces a novel test statistic based on the weighted average of the squared score test statistic (WAST) over the nuisance parametric space. Unlike the aforementioned tests, the proposed WAST has a closed form from the perspective of the conjugate priors when an appropriate weight is chosen.
It is a by-product that the proposed WAST reduces the computational cost dramatically because a closed-form expression is obtained after integrating out the nuisance parameter. Extensive simulation studies with finite samples show that the WAST exhibits 
(i) type-1 errors that are close to the nominal significance level, (ii) higher power compared to its competitors in practice.  
The limiting distributions of the test statistic are established under the null and alternative hypotheses, and the approximation of critical value by the bootstrap method is provided, as are the theoretical guarantees. 


The remainder of this paper is organized as follows. In Section <ref>, we introduce the novel test statistic, WAST, and establish its limiting distributions under the null and alternative hypotheses. In Section <ref>, we investigate the choice of the weight function based on several useful examples that are all special cases of the considered GEEs, with detailed computations included for each example. In Section <ref>, we report the results of simulation studies conducted to evaluate the finite-sample performance of the proposed test statistic and to compare its power with competitors. In Section <ref>, we apply the proposed test method to a real data set. Finally, in Section <ref>, we conclude with remarks and further extensions. The Assumptions are provided in the Appendix, the proofs and additional simulation studies are included in the Supplementary
and in Supplementary C.3, we further extend the proposed test procedure to the GEEs with multiple change planes. An R package named “wast” is available at <https://github.com/xliusufe/wast>.



§ TEST STATISTICS


We first give some useful notation for convenience of expression. For a vector ∈^d and a squared matrix A=(a_ij)∈ℝ^d× d, denote by  the Euclidean norm of , by _p=(∑_i=1^d|v_i|^p)^1/p the p-norm of ; _A^2=∑_i, ja_ijv_iv_j and ^⊗2=. Denote by A_p=sup{A_p: ∈^d, _p=1} the induced operator norm for a matrix A=(a_ij)∈^m× d. 
Denote by  the ordinary probability measure such that f=∫ f d for any measurable function f. Let L^p(Q) be the space of all measurable functions f such that f_Q, p:=(Q|f|^p)^1/p<∞, where p∈[1, ∞) and (Q|f|^p)^1/p denotes the essential supremum when p=∞. Let N(, , ·_Q, 2) be an -covering number of  with respect to the L^2(Q) seminorm ·_Q, 2, where  is a class of measure functions and Q is finite.



 §.§ Preliminaries

We are interested in the testing problem (<ref>).
For any known ∈Θ_θ, we consider a squared score test statistic for testing =, i.e.,

    _n() = n^-1Ψ_n(, , )^2_()^-1,

where  is an estimate of  based on the empirical GEE Ψ_1n()=0 and ()=
n^-1∑_i=1^n{ψ(_i, , , ) - K̂()Ĵψ_1(_i, )}^⊗2. Here, () and  are consistent estimators of K()=∂ψ(, _0, , )/∂∈ℝ^p× r and J=[∂ψ_1(, _0)/∂]^-1∈ℝ^r× r, respectively.


	If Assumptions (A1)–(A5) in the Appendix hold, then for any fixed ∈Θ_θ, _n() converges in distribution to a χ^2 one with p degrees of freedom under H_0 as n →∞.



	Because _n() includes an unknown parameter , Lemma <ref> cannot be used directly in practice. However, it essentially demonstrates that the asymptotic distributions are the same for any nuisance parameter ∈Θ_θ. Thus, the supremum and the weighted average of _n() achieve correct type-1 errors. We construct the proposed test statistic by making use of this point.


Because equations (<ref>) does not depend on  when = 0, the parameter  is identifiable only under the alternative hypothesis. Lemma <ref> suggests that _n() for any ∈Θ_θ is a test statistic that can control type-1 errors. However, _n() may suffer a loss of power for some , making it difficult to determine  in practice. To pursue the largest power theoretically, a good choice is to take the supremum of the squared score test statistic over the parametric space Θ_θ (SST), i.e.,

    _n = sup_∈Θ_θ{n^-1Ψ_n(, , )^2_()^-1}.

The test statistic _n has been widely studied in the literature for the case in which ψ_0(, , ,) is one-dimensional,
and the dimension of the   is small; see <cit.> and <cit.>. We summarize their results under the null and alternative hypotheses in the GEE framework in Lemmas A.4 and A.5 in the Supplementary Material.



 §.§ Weighted average of squared score test statistic

The SST considered in Section <ref> may sacrifice power in practice for pursuing theoretically the largest power, especially for large q and p. To overcome this drawback and take advantage of Lemma <ref>, it is natural to average _n() over Θ_θ. In practice, we can take a weighted average over Θ_θ.

Based on the above arguments, we propose the WAST based on _n() by taking integral after removing both the inverse of covariance and cross-interaction terms:

    T_n = 1/n(n-1)∑_i≠ jρ_ij

with

    ρ_ij = ψ_0(_i, )ψ_0(_j, )∫_∈Θ_θ(_i≥ 0) (_j≥ 0)w()d,

where w() is a weight satisfying w()≥0 for all ∈Θ_θ and ∫_∈Θ_θw()d=1.


Another explanation for w() comes from a Bayesian perspective, in which the weight w() can be treated as the density of the prior of the . Herein, we focus mainly on testing the existence of subgroups instead of estimating the , so there is no requirement to calculate the posterior distribution, which usually makes implementation relatively easy.


In general, there is no closed-form expression for ρ_ij, but it usually has a simple formulation if we choose a specific weight w(). 

The choice of weight can affect the computation of the test statistic since the numerical integration over ^q is required. 
Fortunately, there is a closed-form expression for an appropriate weight from the perspective of the conjugate priors, which is quite attractive in practice. 
<cit.> and <cit.> have discussed a similar weight that facilitates the derivation of the closed-form expression. We discuss the calculation of ρ_ij in detail in Section <ref>.

<cit.> studied the average Lagrange multiplier (LM), denoted by _n = ∫_∈Θ_θ_n()w()d, which is a special case of

    _n=(1+c)^-p/2∫exp(c/2(1+c)_n())dw(),

where c is a scalar constant. However, the main drawback of this approach is that it is hard to find a closed-form expression for _n, which may lose power in practice and incurs a expensive computational cost for large p. By contrast, the proposed WAST T_n has a different form from that of _n, and it is usually easy to achieve a closed-form representation by choosing an appropriate weight w() (see Section <ref> for details).



 §.§ Asymptotic analysis

This subsection is devoted to exploring the asymptotic properties of the proposed test statistic T_n. Before establishing its asymptotic distribution, we introduce some additional notations. Let the kernel of a U-statistic under the null hypothesis be

    h(_i, _j)
    		=   ∫_∈Θ_θψ(_i, _0, , )ψ(_j, _0, , ) w()d
        - ψ_1(_i, _0) K_j- K_iψ_1(_j, _0) + ψ_1(_i, _0) Hψ_1(_j, _0),

where

    H=∫_∈Θ_θJ K() K()J w()d  
    	K_i = ∫_∈Θ_θJ K()ψ(_i, _0, , ) w()d.



	If Assumptions (A1)–(A5) in the Appendix hold, then under the null hypothesis, we have
	
    nT_n -μ_0 ζ,

	where  denotes convergence in distribution, μ_0=-2[ψ_1(_1, _0) K_1]
	+[ψ_1(, _0) Hψ_1(, _0)],
	ζ is a random variable of the form ζ=∑_j=1^∞λ_j(χ^2_1j-1), and χ^2_11, χ^2_12, ⋯ are independent χ^2_1 variables, i.e., ζ has the characteristic function
	
    [e^itζ]=∏_j=1^∞(1-2itλ_j)^-1/2e^-itλ_j.

	Here, i=√(-1) standards for the imaginary unit,
	and {λ_j} are the eigenvalues of the kernel h(_1, _2) under f(, _0, , _0), i.e., they are the solutions of λ_jg_j(_2)=∫_0^∞h(_1, _2)g_j(_1)
 f(_1, _0, , _0)d_1 for nonzero g_j, where f(, , , ) is the density of  with parameters , , and .


Next, we investigate the power performance of the proposed test statistic under the local alternative.

	If Assumptions (A1)–(A6) in the Appendix hold, then under the local alternative hypothesis H_1, that is, = n^-1/2 with a fixed vector ∈Θ_β, we have
	
    nT_n-μ_0 ζ_1,

	where μ_0 is defined in Theorem <ref>, ζ_1 is a random variable of the form ζ_1=∑_j=1^∞λ_j(χ^2_1j(μ_aj)-1), and χ^2_11(μ_a1), χ^2_12(μ_a2), ⋯ are independent noncentral χ^2_1 variables, i.e., ζ_1 has the characteristic function
	
    [e^itζ_1]=∏_j=1^∞(1-2itλ_j)^-1/2exp(-itλ_j+itλ_jμ_aj/1-2itλ_j).

	Here, {λ_j} are the eigenvalues of the kernel h(_1, _2) defined in (<ref>) under f(, _0, , _0), i.e., they are the solutions of λ_jg_j(_2)=∫_0^∞h(_1, _2)g_j(_1)f(_1, _0, , _0)d_1 for nonzero g_j, and each noncentrality parameter of χ^2_1j(μ_aj) is
	
    μ_aj = [ϕ_j(_0)∂log(f(_0, _0, , _0))/∂],    j=1, 2, ⋯,

	where {ϕ_j()} denotes orthonormal eigenfunctions corresponding to the eigenvalues {λ_j}, and _0 
	is generated from the null distribution f(, _0, , _0).


Theorem <ref> implies that the power function of nT_n-μ_0 can be theoretically approximated by F_ζ_1, where F_ζ_1 is the cumulative distribution function of ζ_1. The proof of Theorem <ref> shows that 0<  h(_1, _2)=∑_j=1^∞λ_jμ_aj+o(1) under the local alternative hypothesis. The additional mean ∑_j=1^∞λ_jμ_aj under H_1 can be viewed as a measure of the difference between H_0 and H_1. Because the asymptotic distribution is not common, it is difficult to use in practice; instead, in Section <ref>, we provide a bootstrap method for calculating the critical value or .
Theorems <ref>–<ref> show that the bias μ_0 arises from estimating , which means that μ_0 is zero if  is known. In fact, if  is known, then T_n is exactly a U-statistic with zero mean under the null hypothesis.

We extend the proposed test method to the GEEs with multiple change planes. Due to limit of space, we provide asymptotic distributions of the proposed statistic and simulation results in Appendix C.3 of the Supplementary Material. 



 §.§ Computation of critical value

In Section <ref>, we deduced the asymptotic distributions for the WAST T_n under the null and alternative hypotheses. However, because these asymptotic distributions are not as common as normal, χ^2, and F distributions, it is difficult to calculate the critical value directly. Instead, many related articles recommend the strategy of approximating the limiting null distribution of the test statistic. <cit.> proposed a perturbed test statistic for the SST

    ^*_n=sup_∈Θ_θ1/n∑_i=1^n[ν_i{ψ(_i, , , ) + K̂()Ĵψ_1(_i, )}]^2_()^-1,

where () is defined in (<ref>), and {ν_i,i=1,⋯,n} are n independent and identically distributed copies of a random variable ν generated from 𝒩(0,1) that is independent of .


The critical value C_α can be calculated by the empirical distribution of ^*_n according to replicates, say {_n^*, j,j=1,⋯,N}, where C_α is an upper quantile of the empirical distribution of ^*_n and N is a large integer. The null hypothesis is rejected for an α-level test when _n>C_α, and the  is approximated by N^-1∑_j=1^N(_n>_n^*, j).

For the WAST, we recommend the parametric bootstrap method to approximate the critical value or . For each bootstrap sample {_i^*, b, i=1, ⋯, n}, generated as described in Section 3, where b=1, ⋯, B, we calculate T_n^*, b by

    T_n^*, b = 1/n(n-1)∑_i≠ j∫_∈Θ_θψ(_i^*, b, ^*, b, , )ψ(_j^*, b, ^*, b, , )w()d,

where B is a large integer and ^*, b is the estimate of  under the null hypothesis. The critical value can be approximated by the upper quantile of {T_n^*, b, b=1,⋯,B}. The  is approximated by B^-1∑_b=1^B(T_n>T_n^*, b). In Section <ref>, we discuss the parametric bootstrap method using several useful examples.

If the density f(; , , ) of  is given except for the unknown parameters , , and , and bootstrap samples {_i^*, i=1, ⋯, n} are generated from the distribution f(; , , _0), then Theorem <ref> provides the asymptotic consistency of the bootstrap distribution of the test statistic, where  is the estimate of  under the null hypothesis. 


	If Assumptions (A1)–(A5) and (A7) in the Appendix hold, then under the null hypothesis, we have
	
    sup_x∈|^*(nT_n^*≤ x)-(nT_n≤ x)|  0,

	where  denotes convergence in probability and ^* denotes the probability under the bootstrap procedure.




§ CHOICE OF THE WEIGHT

In Section <ref>, we have theoretically established the asymptotic distributions of T_n under the null and alternative hypotheses in the GEE framework. Here, we discuss in detail the closed form of T_n in practice by taking an appropriate weight. We also consider the bootstrap method for each example.


We first consider the quantile regression in example <ref> of Section <ref>. By Lemma A.1 in the Supplementary Material and taking the weight w() as the multivariate Gaussian density with mean  and covariance Σ, ρ_ij in the WAST (<ref>) reduces to

    ρ_ij=ω_ij_i_j[(Y_i-_i(τ)≤0)-τ][(Y_j-_j(τ)≤0)-τ],

where

    ω_ij = ∫_-∞^_j/_j_Σϕ(z)Φ(1/√(1-ϱ_ij^2)(_i/_i_Σ-ϱ_ij z))dz,     i≠ j,

and ϱ_ij = _iΣ_j(_i_Σ_j_Σ)^-1.

From a Bayesian perspective,  and Σ can be treated as priors and chosen empirically. Although there is no closed-form expression of ω_ij for i≠ j in general, we can approximate ω_ij by the Monte Carlo method. More specifically, we approximate ω_ij by

    ω_ij≈ N^-1∑_k=1^N[Φ(1/√(1-ϱ_ij^2)(_i/_i_Σ-ϱ_ij z_k))(z_k≤_j_Σ^-1_j)],

where z_k follows 𝒩(0, 1) and N is a large integer.
In practice, we can simply choose = and Σ=, whereby Lemma A.1 in the Supplementary Material implies that ω_ij reduces to

    ω_ij=1/4+1/2πarctan(ϱ_ij/√(1-ϱ_ij^2))     i≠ j.


If θ∈[0, 1], then we can take w(θ) as being the density of the beta distribution Beta(λ_1, λ_2) with shape parameters λ_1, λ_2>0. In this case, we find that

    ω_ij = B(min{Z_i, Z_j};λ_1, λ_2),

where B(·;λ_1, λ_2) is the cumulative distribution function of Beta(λ_1, λ_2).
If θ∈ℝ, then we can take w(θ) = ϕ(θ;μ, σ^2) as the density of the Gaussian distribution with mean μ and variance σ^2, which implies that

    ω_ij = Φ(min{Z_i, Z_j};μ, σ^2),

where Φ(·;μ, σ^2) is the cumulative distribution function of 𝒩(μ, σ^2).
Alternatively, we can write =(-θ, 1) and =(1, Z), which imply that (Z≥θ) is equal to (≥0). This allows us to use the bivariate normal density as the weight, which results in ω_ij in (<ref>).


As emphasized in Section 2.2, the choice of the weight affects the computation of the test statistic since the numerical integration over ^q. 
We conduct numerical studies based on GLMs, probit, quantile and semiparametric models to investigate the sensitivity of the weight's choice by comparing ω_ij with the closed-form in (<ref>) and the approximated ω_ij in (<ref>), see details in Appendix B.5 in the Supplemetary Material. From the numerical results, compared with the approximated ω_ij in (<ref>), the test statistic with ω_ij in (<ref>) has higher power uniformly, and only take 10% time computationally when N=10000. This strongly recommends one to use the WAST with the closed-form in (<ref>).

For the bootstrap procedure, as with the wild bootstrap of <cit.>, we independently generate a random variable ν satisfying the following conditions: (i) there are two positive constants c_1 and c_2 satisfying sup{ν∈:ν≤ 0}=-c_1 and inf{ν∈:ν≥ 0}= c_2, where  is the support of ν; (ii) the distribution G of ν satisfies ∫_0^∞ν^-1g(ν)dν=-∫_-∞^0ν^-1g(ν)dν=1/2 and [|ν|]<∞, where g(ν) is the density function of ν; (iii) the τth quantile of ν is zero. We obtain the bootstrapped samples {(Y_i^*, _i, _i, _i), i=1, ⋯, n} and calculate T_n^* given in (<ref>), where Y_i^*=_i(τ)+ν_i|r̃_i| and r̃_i=Y_i-_i(τ), with (τ) being the estimate of (τ) based on Ψ_n()= under the null hypothesis. In practice, one can generate ν following a discrete distribution with (ν=2(1-τ))=1-τ and (ν=-2τ)=τ. 

Consider the GLMs in example <ref>. ρ_ij in the proposed test statistic becomes

    ρ_ij = ω_ij_i_j[Y_i-∂ c(_i)/∂μ_i][Y_j-∂ c(_j)/∂μ_j],

where the weight w() can be chosen as in the case of quantile regression. We generate the bootstrap samples from GLM with the density f(_i, ,,_0), which is the density under the null hypothesis by replacing  with . We obtain the bootstrap samples {(Y_i^*, _i, _i, _i), i=1, ⋯, n} and calculate T_n^* given in (<ref>). The consistency of the test statistic T_n^* is given in Theorem <ref>.

We now consider probit regression model in example <ref>, in which the test statistic is

    ρ_ij = ω_ijψ_0(_i, )ψ_0(_j, ),

where ψ_0(_i, ) = Y_i_iϕ(_i)/Φ(_i) - (1-Y_i)_iϕ(-_i)/Φ(-_i) and ω_ij can be chosen as in GLMs.
The bootstrap samples can be drawn from the probit regression model under the null hypothesis by replacing  with . We obtain the bootstrap samples {(Y_i^*, _i, _i, _i), i=1, ⋯, n} and calculate T_n^* given in (<ref>), where Y_i^*= (ν_i≤_i) and ν_i is independent of _i and normally distributed with zero mean and unit variance. The consistency of the test statistic T_n^* is given in Theorem <ref>.

For the semiparametric models in example <ref>, we have

    ρ_ij=ω_ij[A_i-π(_i, _1)][Y_i-γ(_i, _2)][A_j-π(_j, _1)][Y_j-γ(_j, _2)],

where ω_ij is the same as that in the quantile regression of example <ref>. To calculate the critical value or p-value of T_n, we can use the wild bootstrap strategy to generate bootstrap samples <cit.>. More specifically, we obtain the bootstrap samples as {(Y_i^*, A_i, _i), i=1, ⋯, n} with Y_i^*=γ(_i, _2)+ν_ir̃_i and calculate T_n^* given in (<ref>), where r̃_i=Y_i-γ(_i, _2) and ν_i is independent of (Y_i, A_i, _i) and normally distributed with zero mean and unit variance. 



§ SIMULATION STUDIES




 §.§ Performance for GLMs
 
In this section, we demonstrate the finite-sample performance of the proposed WAST. We report here simulation results from several scenarios involving different numbers of  and distributions of   for GLMs and quantile regression, probit regression and semiparametric models. Due to limit of space, we give simulation results of probit regression and semiparametric models for high- or low-dimensional , as well as multiple change planes, in the Appendix B and C of the Supplementary Material. As mentioned in Section <ref>, we choose Gaussian density with zero mean and covariance  as the weight in all simulations.

For fair comparison, we use the same settings as considered by <cit.>.
Consider the GLM in model (<ref>) with the canonical parameter

    μ =  + (≥0),

where = κ_p and α_2=⋯=α_r=log(1.4). For Gaussian and Poisson families,
we set α_1 = 0.5,  and for binomial family α_1 is chosen so that the proportion of cases in the data set is 1/3 on average under the null hypothesis. θ_2,⋯,θ_q are equally spaced numbers from -1 to 1, and θ_1 is chosen as the negative of the 0.65 percentile of Z_2θ_2+⋯+Z_qθ_q, which means that  divides the population into two groups with 0.35 and 0.65 observations, respectively. For the predictor, we generate (v_2, ⋯,v_max{r,p}, Z_2,⋯,Z_q) from a multivariate normal distribution with mean  and covariance Σ=(ρ̃_ij), where ρ̃_ij=1 if i=j and ρ̃_ij=ρ otherwise. Here, we consider both ρ=0 and ρ=0.3. We set _j=(v_j>0) with j=2,⋯,r, X_k = (v_k>0) with k=2,⋯,p, and _1=1, X_1=1, and Z_1=1.

We evaluate the power under a sequence of alternative models indexed by κ, i.e., H_1^κ:  = κ_p with κ equally spaced in the range (0,κ_max]. We set the sample size as n=(300, 600), and we set 1000 repetitions and 1000 bootstrap samples. For comparison, we consider three methods here: (i) the proposed WAST, (ii) the SST, and (iii) the approximated supremum of likelihood-ratio test statistic (SLRT) given by <cit.>. We calculate both SLRT and SST over {^(k)=(θ^(k)_1,⋯,θ^(k)_q): k=1,⋯,K}. Let ^(k)_-1=^(k)_-1/^(k)_-1, where ^(k)_-1=(θ^(k)_2,⋯,θ^(k)_q) and ^(k)_-1=(θ̃^(k)_2,⋯,θ̃^(k)_q), and ^(k)_-1 is drawn independently from the multivariate normal distribution 𝒩(_q-1,_q-1). For each θ_1^(k), k=1,⋯,K, we select it by an equal grid search in the range from the lower tenth percentile to the upper tenth percentile of the data points of {θ^(k)_2Z_i2+⋯+θ^(k)_qZ_iq}_i=1^n, which is the same as that in <cit.>. Here, we set K=1000. 
And the critical value of the test statistic SST is calculated with the resampling method used in <cit.>. The results for each scenario are summarized based on 1000 repetitions.

<ref> lists the type-1 errors (κ=0) with the nominal significance level of 0.05. As can be seen, the sizes of the proposed method are close to 0.05. For the SST, its sizes are far from 0.05 in most of the scenarios, and it performs the worst of these three methods. For the SLRT, its sizes are close to 0.05. The type-1 errors of the Gaussian and Poisson families and other settings for different parameter dimensions can be found in Table C.1 in the Supplementary Material, from which we can observe that the proposed method WAST has a good control over the type-1 error rate.

<ref> and Figures C.1-C.3 in the Appendix C.1 of the Supplementary Material show the powers for all the scenarios considered in <ref> and Table C.1. As can be seen, the powers increase as the sample size n increases, which is verified by the asymptotic theory. For the Gaussian, binomial and Poisson families, the power of the proposed WAST increases faster than those of the SST and SLRT, and this superiority is particularly evident when n is small or p and q increase. In summary, the proposed test statistic shows very competitive performance.

We consider different distributions of grouping variable  in Appendix C.2 of the Supplementary Material. From Tables C.2-C.3 and Figures C.4-C.9, we 
have similar conclusion as these for GLMs with  generated from a normal distribution with zero mean and unit variance. 
For large numbers of , we analyze numerically performance of the WAST in Appendices B.3-B.4 of the Supplementary Material, from which  the WAST still overperforms the competitors 
as scenarios with small numbers of grouping variables.







 §.§ Performance for quantile regression

For fair comparison, we consider the settings as similar to the existing literature <cit.>, which are different from the GLMs in Section <ref>.

Consider the quantile regression model (<ref>),

    Y_i = 0.5+_i+_i(τ)(_i(τ)≥0) + _i,

where _i is generated from standard normal distribution, _i from p-variate normal distribution with mean  and variance √(2)I, and Z_1i=1 and (Z_2i,⋯,Z_qi) from (q-1)-variate standard normal distribution. The error _i is generated from standard normal distribution. 
We set =(1,⋯,1) under H_0, and (θ_2, ⋯,θ_q) = (1,2,⋯,2) under H_1, where θ_1 is chosen as the negative of the 0.65 percentile of Z_2θ_2+⋯+Z_qθ_q, which means that  divides the population into two groups with 0.35 and 0.65 observations, respectively.

For all models with change plane analysis, we evaluate the power under a sequence of alternative models indexed by κ, that is H_1^κ: ^κ=κ^* with κ = i/10 for semiparametric model and κ=i/20 for others, i=1,⋯,10, where ^*=(1,⋯,1). We set sample size n=(200, 600), 1000 repetitions and 1000 bootstrap samples, and report in <ref> and Figures C.11-C.12 in the Appendices C.4 of the Supplementary Material the performance for both the WAST and SST. We calculate SST over {^(k)=(θ^(k)_1,⋯,θ^(k)_q): k=1,⋯,K} with the number of threshold values K=2000. Let ^(k)_-1=^(k)_-1/^(k)_-1, where ^(k)_-1=(θ^(k)_2,⋯,θ^(k)_q) and ^(k)_-1=(θ̃^(k)_2,⋯,θ̃^(k)_q), and ^(k)_-1 is drawn independently from (r-1)-variate standard normal distribution.  For each θ_1^(k), k=1,⋯,K, we select it by equal grid search in the range from the lower 10th percentile to upper 10th percentile of the data points of {θ^(k)_2Z_2i+⋯+θ^(k)_qZ_qi}_i=1^n, which is same as that in <cit.>. Here we consider two combinations of (p,q)=(1, 3), (5, 5). The power results of another two combinations of (p,q)= (10, 10), (50, 20) can be found in Figures C.13-C.14 in Appendix C.4 of the Supplementary Material.

Type-1 errors (κ=0) with sample size n=(200, 600) are listed in <ref> and Table C.5 in Appendix C.4 of the Supplementary Material. We can see from these tables that the size of the proposed WAST are close to the nominal significance level 0.05, but for most scenarios the size of the SST are much smaller than 0.05.
<ref> and Figures C.11-C.14 in the Appendix C.4 of the Supplementary Material indicate that powers become greater as sample size n increases, which are verified by the asymptotic theory. The WAST has comparable power with 
the SST for the quantile regression, but the size of the SST is much less than the nominal level 0.05 in <ref>. To save space here, we report the performance of the WAST 
for probit and semiparametric models in Appendix C.4 of the Supplementary Material, 
and the performance of the WAST for the large numbers of  is analysed in Appendices B.1-B.2 of the
Supplementary Material. We can observe that the WAST works as well in scenarios with a large number of grouping variables as it does in situations with a small number of grouping variables. 







§ EMPIRICAL APPLICATION





In this section, we demonstrate our proposed method by analyzing a data set obtained from China Health and Retirement Longitudinal Study (CHARLS) in 2013. This nationally representative data set focuses on individuals aged 45 and above in China, and includes comprehensive demographic, health, and economic information. CHARLS aims to investigate the primary health and economic adaptations associated with the country's rapid population aging. 
It is well known that the pension income from the society and financial support from adult children are two main sources of income for the elderly. 
Our analysis aims to determine if there are specific subgroups in which pension income affects the elderly's support income differently. The subjects of our study are rural seniors aged 60 and older in Shanxi Province. After data-cleaning, the data set in Shanxi Province includes information from 183 individuals.

We treat the support income (1000 CNY/year) for the elderly as the response variable Y. The baseline variable  includes 
pension income (10,000 CNY/year), the number of children with an annual income of more than 10,000 CNY, and emotional comfort (whether the elderly contact with their children at least every half month: Yes = 1, No = 0). 
To identify the different effects of pension income on support income is of particular interest, therefore, the grouping difference variable  is taken as the pension income. 
The grouping variables  contain some individual characteristics of elderly people, including gender (male = 1, female = 0), age, and living arrangement (whether they live with children: Yes = 1, No = 0).

Figure D.1 in the Supplementary Material exhibits the histogram of support incomes for the elderly in Shanxi Province. 
It can be noticed that the support income has a heavy-tailed distribution. 
In view of this, we consider quantile change-plane regression for different τ∈ (0,1):

    Q_Y_i(τ|_i, _i, _i) = _i(τ)+_i(τ)(_i(τ)≥0),

where Y_i, _i, _i, _i are sample values of Y, , ,, respectively, and the first element of _i is 1 which corresponds to the intercept.

Calculated based on 5000 bootstrap samples, the p-values of the proposed WAST are 0.021 if τ=0.25, <0.0001 if τ=0.5, and 0.265 if τ=0.75. 
This demonstrates that for those who have low and median support incomes, we have strong evidence for rejecting H_0, i.e., there exist two subgroups wherein the pension income has a different impact on support income of the elderly. 
Furthermore, it suggests that the pension income does not have a subgroup effect on the support income for individuals with high support incomes.

Next, we estimate parameters (,,) according to the test results. We first randomly choose 1000 s.  For every given , 
the estimator of  and  can be obtained from classic quantile regression. 
Then the estimated grouping parameter  is obtained by minimizing the empirical quantile loss functions over the given 1000 s, and consequently, one of subgroups is given by {i:_i≥0}.

<ref> lists the estimators for τ=0.25 and 0.5. According to  in <ref>, one of subgroups, denoted by G^+={i:^T_i(τ)≥ 0}, is 
    G^+={i:-0.051*gender+0.014*age-0.023*living  arrangement≥ 0.998}.
 There are 50 individuals in the subgroup G^+. Based on these results, we can see that for elderly people in the subgroup G^+, the pension income has a positive impact on the support income. 
Conversely, the effect is reversed in another subgroup. 



All elderly people aged over 77 years fall within the estimated subgroup G^+, and thus their support income will increase as their pension income increases. 
There may be two reasons for this. Firstly, the living expenses of these individuals may be higher. Therefore, their pension income may be insufficient to cover their cost of living, necessitating the need for additional support income. 
Secondly, pension income can increase the unearned income of the elderly, allowing them to reduce working hours, have more leisure time, and consequently increase intergenerational care time. As a result, children may increase their financial support for their parents as a form of reciprocity. 

The estimator is =(1.019, -0.288, 2.125, 0.694) when τ=0.75, which indicates that for individuals with high support income, the pension income exerts a negative impact on the support income.



§ CONCLUSION


Herein, we have considered a class of change-plane tests for heterogeneous estimating equations. A novel test statistic, WAST, is introduced to test whether the change plane exists. In a comparison with existing test procedures, including the supremum of the squared score test from the framework of the average exponential LM test <cit.>, the proposed test statistic significantly improves the power in practice, particularly when the number of grouping variables is large. As a by-product, the WAST reduces the computational cost dramatically because it has a closed form when an appropriate weight is used. Asymptotic distributions are derived under the null and alternative hypotheses, and we extend the proposed method to models with multiple change planes in the GEE framework, as studied by <cit.>.
Many classic and useful models in the GEE framework are considered herein, such as quantile regression <cit.>, GLMs <cit.>, probit regression <cit.>, and semiparametric models <cit.>. Case studies are analyzed to further demonstrate the performance of the proposed test procedure. From the simulation studies and case studies, we conclude that compared with several competitors, the proposed WAST performs pretty well not only for low-dimensional scenarios but also for situations involving a large number of grouping variables.

We provide the hypothesis test to detect the existence of subgroups, but it is left blank how to partition the subjects, particularly in the setting with large numbers of grouping variables, which needs the estimate of parameters, especially of the grouping parameter . It is a possible strategy for the estimation of the grouping parameter to penalize the loss function with _1 under the assumption of sparsity.



§ ACKNOWLEDGEMENTS

This work was supported by National Natural Science Foundation of China (12271329) and Program for Innovative Research Team of SUFE.



§ DISCLOSURE STATEMENT

The authors report there are no competing interests to declare.



§ SUPPLEMENTARY MATERIAL

Appendix A includes proofs of Theorems <ref>–<ref>
and related Lemmas. Appendices B and C present additional simulation studies to illustrate the performance of the proposed test statistic. 
Appendix D shows the histogram of the support incomes for the elderly in Shanxi Province and additional analysis results of another real dataset.



§ APPENDIX

We, herein, provide some assumptions required to establish the asymptotic properties.

	
  * ψ_1(, _0)=0 for some _0∈Θ_α. ψ_1(, )→ implies that -_0=o(1) for any sequence {}∈Θ_α.
	
  * The maps ↦ψ(, , , ) and ↦ψ_1(, ) are twice continuously differentiable for any ∈Θ_β and ∈Θ_θ. 
	The map ↦ψ(, _0, , ) is continuously differentiable for any ∈Θ_θ and ∫_∈Θ_θ∂ψ(, _0, , )/∂ w()d=O(1).
	
  * max_j=1, ⋯, p[ψ^(j)(, , , )-ψ^(j)(, ', , )]^2≤ L-'^2 for any ∈Θ_α, '∈Θ_α, and ∈Θ_θ, where ψ^(j)(, , , ) is the jth component of ψ(, , , ) and L is a constant. max_j=1, ⋯, r[ψ_1^(j)(, )-ψ_1^(j)(, ')]^2≤ L_1-'^2 for any ∈Θ_α, and '∈Θ_α, where ψ_1^(j)(, ) is the jth component of ψ_1(, ) and L_1 is a constant.
	
  * The classes of functions ^(j)={↦ψ^(j)(, , , ): ∈Θ_α, ∈Θ_β, ∈Θ_θ} and _1^(j)={↦ψ_1^(j)(, ): ∈Θ_α} are pointwise measurable and satisfy the uniform entropy condition max_j=1, ⋯, plogsup_Q N(F_j_Q, 2, ^(j), ·_Q, 2) ≤ C_0log(e/), where F_j=sup_f∈^(j)|f|, ∈(0, 1] and C_0≥1 is a constant. The classes ^(j) and _1^(j) have measurable envelope functions F_j≥sup_f∈^(j)|f| satisfying F_j^s()≤ C and F_1j≥sup_f∈_1^(j)|f| satisfying F_1j^s()≤ C_1, respectively, where s≥ 4.
	
	
  * J=[∂ψ_1(, _0)/∂]^-1∈ℝ^r× r is a finite and positive-definite deterministic matrix, and sup_∈Θ_θ, ∈Θ_βK(, )_∞≤ C_K, J_2≤ C_c, where
	K(, )=∂ψ(, _0, , )/∂∈ℝ^p× r is continuous with respect to ∈Θ_β. Denote K()=K(, ). ψ_1(, )^2 and ψ(, , , )^2 are bounded for any ∈𝒰_α_0 and ∈Θ_θ, where 𝒰_α_0 is a neighborhood of _0.
	
	
	
  * There is a positive function b(, ) of  relying on _0, _0 such that
	
    |∂ f(; _0, r_n, _0)∂/f(; _0, , _0)|≤ b(, ),

	and [b(, )^2], λ_max([b(, )ψ_1(, _0)^⊗2]), and λ_max([b(, )ψ(, _0,,)^⊗2]) 
	are bounded by C_, and for all k, [ϕ_k()^2b(, )] is bounded by C_, where ∈Θ_β, r_n=o(1), C_>0 is a constant relying on , ϕ_k(·) is defined in Theorem <ref>, and  is generated from the null distribution with density f(; _0, , _0).
	
	
  * There is a positive function b_1(, _1) of  relying on _0, _0 such that
	
    |_1∂ f(; _0+r_n_1, , _0)/∂/f(; _0, , _0)|≤ b_1(, _1),

	and [b_1(, _1)^2], λ_max([b_1(, _1)ψ_1(, _0)^⊗2]), 
	and λ_max([b_1(, )ψ(, _0,,)^⊗2])
	are bounded by C_f(_1), and for all j, k, [ϕ_k()^2b_1(, _1)] and [|ϕ_k()ϕ_j()|b_1(, _1)] are bounded by C_f(_1), where _1∈Θ_α, r_n=o(1), C_f(_1)>0 is a constant relying on _1, ϕ_k(·) is defined in Theorem <ref>, and  is generated from the null distribution with density f(; _0, , _0). Assume that, for _2∈Θ_α and any ∈Θ_α satisfying -_0^2=O(n^-1),
	
    [(_2∂ h(_1, _2, )/∂)^2(1+b_1(_1, _1))(1+b_1(_2, _1))]
    	<∞,

	where h(_1, _2, ) equals h(_1, _2) defined in (<ref>) with _0 replaced by , and _1 and _2 are independently generated from the null distribution with density f(, _0, , _0).



Assumptions (A1)–(A5) are mild conditions for establishing the asymptotic properties of statistic _n. Assumption (A1) is an identifiability condition for the GEE; see Theorem 2.10 of <cit.>. Assumptions (A2) and (A3) are imposed for the asymptotic properties of the estimate of  and the test statistic _n. Assumption (A4) involves entropy conditions and is analogous to the assumption imposed by <cit.>. Assumption (A5) is mild and easily verified in practice, and is used to derive the asymptotic distribution of the proposed statistic T_n.
Assumptions (A6) and (A7) basically impose the Lipschitz condition, which is similar to the Lipschitz condition in Assumption (A2).



A.equationAsection

Appendix section



§ PROOF OF THEOREMS

A.lem

We first give some useful notation for convenience of expression. For a vector ∈^d and a squared matrix A=(a_ij)∈ℝ^d× d, denote by  the Euclidean norm of , by (A)=∑_i=1^da_ii the trace of A; _A^2=∑_i, ja_ijv_iv_j and ^⊗2=. Denote by A_p=sup{A_p: ∈^d, _p=1} the induced operator norm for a matrix A=(a_ij)∈^m× d.

Denote by  the ordinary probability measure such that f=∫ f d for any measurable function f, by  the empirical measure of a sample of random elements from  such that f=n^-1∑_i=1^nf(_i), and by  the empirical process indexed by a class  of measurable functions such that f=√(n)(-)f for any f∈. Let L^p(Q) be the space of all measurable functions f such that f_Q, p:=(Q|f|^p)^1/p<∞, where p∈[1, ∞) and (Q|f|^p)^1/p denotes the essential supremum when p=∞. Let N(, , ·_Q, 2) be an -covering number of  with respect to the L^2(Q) seminorm ·_Q, 2, where  is a class of measure functions and Q is finite discrete.


	Let  be normally distributed with mean  and covariance Σ. Denote by ϕ(z) and Φ(z) the density function and cumulative distribution function of the standard normal distribution, respectively. For any nonzero vectors ∈ℝ^p and ∈ℝ^p and any numbers η_a∈ℝ and η_b∈ℝ, we have
	
    [(≥η_a)]=   Φ(-η_a/(Σ)^1/2),
    [(≥η_a)(≥η_b)]=   ∫_η_b-/(Σ)^1/2^+∞ϕ(z)Φ(1/√(1-ρ^2)(-η_a/(Σ)^1/2+ρ z))dz,

	where ρ = Σ
/(ΣΣ)^1/2, and (·) denotes the indicator function.
	Moreover, when -η_a=-η_b=0, we have
	
    [(≥η_a)]=   1/2,
    [(≥η_a)(≥η_b)]=   1/4+1/2πarctan(ρ/√(1-ρ^2)).
	

Proof of Lemma <ref>. Let μ_1 =, μ_2 =, σ_1^2 = Σ, σ_2^2 = Σ.
Noting that the random variables Z_1 = -μ_1 ∼ N(0, σ^2_1) and Z_2 = -μ_2∼ N(0, σ^2_2), we have Z=(Z_1,Z_2)∼ N(, Γ) with

    Γ = (
    	[ Σ Σ; Σ Σ ]),

which implies that 

    [(≥η_a)]
    	=   [(Z_1≥η_a-μ_1)]
    
    	=   ∫_η_a-μ_1^+∞1/√(2π)σ_1exp{-z_1^2/2σ_1^2}dz_1
    
    	=   Φ(μ_1-η_a/σ_1),

and

    [(≥η_a)(≥η_b)] 
    
    	=   [(Z_1≥η_a-μ_1)(Z_2≥η_b-μ_2)] 
    
    	=   ∫_η_b-μ_2^+∞∫_η_a-μ_1^+∞1/2πσ_1σ_2√(1-ρ^2)exp{-1/2ZΓ^-1Z}dz_1dz_2
    
    	=   ∫_η_b-μ_2^+∞∫_η_a-μ_1^+∞1/2πσ_1σ_2√(1-ρ^2)exp{-1/2(1-ρ^2)(z_1/σ_1-ρz_2/σ_2)^2-z_2^2/2σ_2^2}dz_1dz_2
    
    	=   ∫_η_b-μ_2^+∞1/√(2π)σ_2exp{-z_2^2/2σ_2^2}Φ(-1/√(1-ρ^2)(η_a-μ_1/σ_1-ρ/σ_2z_2))dz_2
    
    	=   ∫_η_b-μ_2/σ_2^+∞ϕ(z)Φ(1/√(1-ρ^2)(μ_1-η_a/σ_1+ρ z))dz.


When -η_a=-η_b=0, it is straightforward that

    [(≥η_a)(≥η_b)]
    	=   ∫_0^+∞ϕ(z)Φ(ρ/√(1-ρ^2)z)dz
    
    	=   1/4+1/2πarctan(ρ/√(1-ρ^2)).
 □


	If Assumptions (A1)-(A4) hold, we have
	
    -_0
    		=    - Jψ_1(, _0)+o_p(n^-1/2),

	where _0 is given in Assumption (A1), and J=(∂ψ_1(, _0)/∂)^-1∈ℝ^r× r.	

Proof of Lemma <ref>.
Note that

    [ψ_1(, )- ψ_1(, _0)]
        =   ψ_1(, ) -ψ_1(, _0)
       - (-)[ψ_1(, )-ψ_1(, _0)].

It suffices to prove that

    (-)[ψ_1(, )-ψ_1(, _0)] = o_p(n^-1/2),

and

    [ψ_1(, )- ψ_1(, _0)] = J^-1(-_0)+o(n^-1/2).

	
	Let 𝒰_α_0={∈Θ_α:-_0=o(1)} and _1α=∪_1≤ j≤ r_1α^(j), where _1α^(j)={ψ_1^(j)(, )-ψ_1^(j)(, _0):∈𝒰_α_0}.
        By Assumptions (A3)-(A4) and Theorem 5.1 and 5.2 of <cit.> with σ^2=F_1α_,2^2 and M_n=max_1≤ i ≤ nF_1α(_i), we have that, with probability 1-o(1), 
	
    sup_f∈_1α f≲ 2√(C_0r)L_1^1/2-_0+4C_0rM_nn^-1/2=o(1),

	which implies that (<ref>) holds.
	
	
	It follows from the central limit theorem
	
    ψ_1(, _0) = O_p(n^-1/2),

	which combining (<ref>) indicates that [ψ_1(, )- ψ_1(, _0)] = O(n^-1/2) with probability 1-o(1), and consequently, we have, by Assumption (A1), -_0=o(1) with probability 1-o(1).
	
	For any ∈𝒰_α_0, we have, by Assumption (A2) and Taylor's theorem,
	
    [ψ_1(, )- ψ_1(, _0)]
    		= ∂ψ_1(, _0)/∂(-_0) + O(-_0^2),

	which indicates that, with probability 1-o(1),
	
    [ψ_1(, )- ψ_1(, _0)]
    		= J^-1(-_0)(1+o(1)).

	Thus, combining (<ref>) and (<ref>) we have -_0=O_p(n^-1/2), and that (<ref>) holds. □

    
        If Assumptions (A1)–(A5) hold, then for any ∈Θ_β and ∈Θ_θ, we have
        
    ψ(, , , )
            =   ψ(, _0, , ) - K(, )Jψ_1(, _0)+o_p(n^-1/2),

        where _0 is given in Assumption (A1), K(, )=∂ψ(, _0, , )/∂∈ℝ^p× r, and J={∂ψ_1(, _0)/∂}^-1∈ℝ^r× r.
        
        Proof of Lemma <ref>.
            Note that
            
    [ψ(, , , )- ψ(, _0, , )] = ψ(, , , ) -ψ(, _0, , )
    
                    - (-)[ψ(, , , )-ψ(, _0, , )].

            It suffices to prove that
            
    (-)[ψ(, , , )-ψ(, _0, , )] = o_p(n^-1/2),

            and
            
    [ψ(, , , )- ψ(, _0, , )] = - K(, )Jψ_1(, _0)+o(n^-1/2).

            
            We first show that (<ref>) holds.
            Let 𝒰_α_0={∈Θ_α:-_0=O(n^-1/2)} and _α=∪_1≤ j≤ p_α^(j), where _α^(j)={ψ^(j)(, , , )-ψ^(j)(, _0, , ):∈𝒰_α_0,∈Θ_β, ∈Θ_θ}. We define the envelope function of _α as
            
    F_α = max_j=1,⋯,pF_α^(j),

            where
            
    F_α^(j) = sup_∈𝒰_α_0, ∈Θ_θ|ψ^(j)(, , , )-ψ^(j)(, _0, , )|.

            Assumption (A3) implies that
            
    F_α_, 2≤F_,2≤ L^1/2-_0,

            and by the fact that _α^(j)⊆^(j)-^(j),
            
    logsup_Q N(F_α_Q,2, _α, ·_Q,2)
                    ≤    2p max_j=1,…,plogsup_Q N(/2F_j_Q,2, ^(j), ·_Q,2)
    ≤    2C_0plog(2e/),

            which results in that, for δ∈ (0,1),
            
    (δ,_α, F_α)
                    =   ∫_0^δsup_Q√(1+log N(F_α_Q,2, _α, ·_Q,2))d
    ≤   ∫_0^δ√(1+2C_0plog(2e/))d
    
                    =   2e∫_2e/δ^∞1/^2√(1+2C_0plog())d.

            According to the fact that 1+log()≥1 for all ∈[1, ∞) and
            the integration by parts, we have
            
    ∫_2e/δ^∞1/^2√(1+log())d
                    =   -1/√(1+log())|_=2e/δ^∞+1/2∫_2e/δ^∞1/^2√(1+log())d
    ≤   δ/2e√(1+log(2e/δ))+1/2∫_2e/δ^∞1/^2√(1+log())d,

            which implies that
            
    ∫_2e/δ^∞1/^2√(1+log())d≤δ/e√(1+log(2e/δ)),

            and consequently
            
    (δ,_α, F_α) ≤δ√(2C_0p(1+log(2e/δ))).

            Therefore, by Assumption (A4) and Theorem 5.1 and 5.2 of <cit.> with σ^2=F_α_,2^2 and M_n=max_1≤ i ≤ nF_α(_i), we have that, with probability 1-o(1),
            
    sup_f∈_α f≲ 2√(C_0p)L^1/2-_0+4C_0pM_nn^-1/2=o(1),

            which implies that, with probability 1-o(1),
            
    sup_f∈_α(-)f = o(n^-1/2)

            because of the fact that ∈𝒰_α_0 with probability 1-o(1) by Lemma <ref>, and consequently (<ref>) holds.
            
            We, now show (<ref>) holds. For any ∈𝒰_α_0, we have, by Assumption (A2) and Taylor's theorem,
            
    [ψ(, , , )- ψ(, _0, , )]
                    = ∂ψ(, _0, , )/∂(-_0) + O(-_0^2),

            which combining Lemma <ref> imply that, with probability 1-o(1),
            
    [ψ(, , , )- ψ(, _0, , )]
                    = - K(,)Jψ_1(,α_0) + O(n^-1),

            and consequently (<ref>) holds. □

            Proof of Lemma 1.
By applying the central limit theorem to i.i.d. sample {_i}_i=1^n, we have that under H_0
	
    n^1/2[ψ(, _0, , )	- K()Jψ_1(, _0)] N(, Γ()),

	for any fixed ∈Θ_θ, where Γ()={ψ(, _0, , ) - K()Jψ_1(, _0)}^⊗2. 
 By Lemma <ref>, 
 it is straightforward that for any fixed ∈Θ_θ,
	
    _n()
    		= ψ(, _0, , ) - K()Jψ_1(, _0)_()^-1^2+o_p(1)
    		χ_p^2.

	To complete the proof, it suffices to show that ()Γ() for any fixed ∈Θ_θ. In fact, it is easy to show that ()Γ() for any fixed ∈Θ_θ by the law of large numbers.	□

    
        If Assumptions (A1)–(A5) hold, then _n converges in distribution to sup_∈Θ_θ^2() under H_0 as n →∞, where {(); ∈Θ_θ} is a zero-mean Gaussian process with the covariance function
        
    Σ(_1, _2) = ψ_*(, _0, , _1)ψ_*(, _0, , _2),

        indexed by _1∈Θ_θ and _2∈Θ_θ, where ψ_*(, _0, , )=Γ()^-1/2{ψ(, _0, , ) - K()Jψ_1(, _0)}
        and Γ()={ψ(, _0, , ) - K()Jψ_1(, _0)}^⊗2, with K() and J defined in Assumption (A5).
        
        Proof of Lemma <ref>.
        
            Define a class of measurable functions _2=∪_1≤ j≤ p_2^(j) and its envelope function F_2=max_j=1,⋯,psup_f∈_2^(j)|f|, where _2^(j)={↦ψ_*^(j)(, _0, , ): ∈Θ_β, ∈Θ_θ}. Assumption (A2) and (A4) imply that
            
    F_2^2()≤ (C+C_K^2C_c^2C_1).

            and by the fact that _2^(j)⊆^(j)+_1^(j)+⋯+_1^(j)_r, and
            
    logsup_Q N(F_2_Q,2, _2, ·_Q,2)
                    ≤    pmax_j=1,…,plogsup_Q N(/2F_j_Q,2, ^(j), ·_Q,2)
        +rpmax_j=1,…,plogsup_Q N(/2F_1j_Q,2, _1^(j), ·_Q,2)
    ≤    2C_0rplog(2e/),

            which results in that, 
         along the line of Lemma <ref>, 
         for δ∈ (0,1),
            
    (δ,_2, F_2)
                    =   ∫_0^δsup_Q√(1+log N(F_2_Q,2, _2, ·_Q,2))d
    ≤   δ√(2C_0rp(1+log(2e/δ)))< ∞.

            By Theorem 2.5.2 of <cit.>, _2 is -Donsker, which means that ψ_*(, _0, , ) converges weekly to a mean zero Gaussian process with the covariance
            
    Σ(_1, _2) = ψ_*(, _0, , _1)ψ_*(, _0, , _2).

         By Lemma <ref>, 
         it remains to show that ()Γ() for any fixed ∈Θ_θ. In fact, it is seen that ()Γ() for any fixed ∈Θ_θ by the law of large numbers. □
   
         
            If Assumptions (A1)–(A5) hold, then _n converges in distribution to sup_∈Θ_θ_r_n^2() under H_1n as n →∞, where {_r_n(); ∈Θ_θ} is a Gaussian process with the mean function
            
    μ() = Γ()^-1/2∂_0ψ(, _0, , )/∂

            and the covariance function Σ(_1, _2) as defined in Lemma <ref>. Here, _0 is the probability measure under the null hypothesis.
            
            Proof of Lemma <ref>.
            
                Under the local alternative, as in Lemma <ref>, we can show that
                
    (-)[ψ(, , , )-ψ(, _0, , )] = o_p(n^-1/2),

                and consequently,
                
    ψ(, , , )
                    =   ψ(, _0, , )
       +[ψ(, , , )-ψ(, _0, , )]+o_p(n^-1/2).

                By the differentiability of ψ(, , , ) with respect to  in Assumption (A2), we have, for any ∈Θ_α and ∈Θ_θ,
                
    ψ(, , , ) =ψ(, , , )+∂ψ(, , , )/∂ + O(^2).

                Thus, for any  satisfying -_0=O(n^-1/2), we have, by Assumption (A2) and Taylor's theorem,
                
    [ψ(, , , )- ψ(, _0, , )]
                        =   ψ(, , , ) -∂ψ(, , , )/∂
       -ψ(, _0, , )+∂ψ(, _0, , )/∂+ O(^2)
    
                        =    K(,)(-_0)+∂^2[ψ(, _0, , )/∂∂(-_0)
       + O(-_0^2)+O(^2),

                which combining Lemma <ref> implies that, with probability 1-o(1),
                
    [ψ(, , , )- ψ(, _0, , )]
                        =    K(,)Jψ_1(,α_0) + O(n^-1),

                which implies that
                
    ψ(, , , )
                        =   {ψ(, _0, , )+ K(,)Jψ_1(, _0)}+o_p(n^-1/2)
    
                        =   {ψ(, _0, , )+ K(,)Jψ_1(, _0)}
       +{ψ(, _0, , )-ψ(, _0, , )}+o_p(n^-1/2).

                Let _0 denote the probability measure under the null. Since under the local alternative,
                
    μ()
                    =   Γ()^-1/2∂_0ψ(, _0, , )/∂,

                we have, along the line of Lemma <ref>, 
                ψ_*(, _0, , ) converges weekly to a Gaussian process with mean function μ() and the covariance
                
    Σ(_1, _2) = ψ_*(, _0, , _1)ψ_*(, _0, , _2).

                Therefore, we complete the proof by (<ref>). □

                
                    Let 
    H = ∫_∈Θ_θJ K() K()J w() d,

                    where J and K() are defined in Assumption (A5), and w() is a weight satisfying w()≥0 for all ∈Θ_θ and ∫_∈Θ_θ w()d=1.
                    If Assumption (A5) holds, we have
                    
    λ_max(H)≤λ_max^2(J)λ_max[∫_∈Θ_θΣ_K() w() d],

                    and
                    
    (H)≤λ_max^2(J)(∫_∈Θ_θΣ_K() w() d),

                    where Σ_K()=K() K().
                    Furthermore, by Assumption (A5), λ_max(H) and (H) are bounded.
                
                Proof of Lemma <ref>. It is straightforward to complete the proof by utilizing Assumption (A5). □
 
                
                    If Assumption (A4) holds, we have for any fixed ∈Θ_β,
                    
    [h(_1,_2)^2]	≤   4∫_∈Θ_θλ_max(Σ_ψ())(Σ_ψ())w()d
       +4λ_max^2(Σ_ψ_1)(H^2)
       +8λ_max(Σ_ψ_1)λ_max^2(J)∫_∈Θ_θλ_max(Σ_ψ)(Σ_K())w()d,

                    where h(_1,_2) is defined in (11) in the main paper, and _1 and _2 are generated from the null distribution with density f(,_0,,_0) and are independent of each other.
                    Furthermore, if Assumption (A5) holds, [h(_1,_2)^2] is bounded.
                    
                    Here we define matrices as following,
                    
    Σ_ψ()=   [ψ(, _0, , )^⊗2],
    Σ_ψ_1()=   [ψ_1(, _0)^⊗2],
    Σ_K()=   K() K().

                
                Proof of Lemma <ref>. We deal with each term in h(_1,_2) separately. It is seen that
                
    [(∫_∈Θ_θψ(_i, _0, ,)ψ(_j, _0, ,) w()d)^2]
    ≤   ∫_∈Θ_θ[(ψ(_i, _0, ,)ψ(_j, _0, ,))^2] w()d
    ≤   ∫_∈Θ_θλ_max(Σ_ψ())[ψ(_i, _0, ,)^2]w()d
    ≤   ∫_∈Θ_θλ_max(Σ_ψ())(Σ_ψ())w()d.

                By the definition of K_i, we have
                
    K_i^2
                        ≤   ∫_∈Θ_θ{[ J K()ψ(_i, _0, ,)ψ(_i, _0, ,) K()J]} w()d
    ≤   ∫_∈Θ_θλ_max(Σ_ψ())(J K() K()J) w()d
    
                        =   λ_max^2(J)∫_∈Θ_θλ_max(Σ_ψ)(Σ_K())w()d,

                and consequently,
                
    [(ψ_1(_i, _0) K_j)^2]
                        =   [(K_jΣ_ψ_1K_j)] 
    ≤   λ_max(Σ_ψ_1)λ_max^2(J)∫_∈Θ_θλ_max(Σ_ψ)(Σ_K())w()d.

                And similarly, it can be obtained that
                
    [(ψ_1(_i, _0) H ψ_1(_j, _0))^2]
                        ≤   λ_max^2(Σ_ψ_1)(H^2).

                From the foregoing considerations, it is seen that
                
    [h(_1,_2)^2]
                        ≤   4_ij(_0)+4[ψ_1(_i, _0) Hψ_1(_j, _0)]^2
                        +8[ψ_1(_i, _0) K_j]^2
    ≤   4∫_∈Θ_θλ_max(Σ_ψ())(Σ_ψ())w()d
       +4λ_max^2(Σ_ψ_1)(H^2)
       +8λ_max(Σ_ψ_1)λ_max^2(J)∫_∈Θ_θλ_max(Σ_ψ)(Σ_K())w()d.

                By Assumption (A5) and Lemma <ref>, it is straightforward to see that [h(_1,_2)^2] is bounded. □
 
                
                    If Assumption (A4) – (A6) holds, we have for any fixed ∈Θ_β,
                    
    [h(_1,_2)^2(1+b(_1,))(1+b(_2,))]

                    is bounded,
                    where b(,) is defined in Assumption (A6), h(_1,_2) is defined in (11) in the main paper, and _1 and _2 are generated from the null distribution with density f(,_0,,_0) and are independent of each other.
                    
                
                Proof of Lemma <ref>. 	We defined Σ_ψ,b and Σ_ψ_1,b as following,
                
    Σ_ψ,b=   [b(_i,)ψ(, _0, , )^⊗2],
    Σ_ψ_1,b=   [b(_i,)ψ_1(_i, _0)^⊗2].

                Along the lines of Lemma <ref>, we have by Assumption (A6),
                
    [(∫_∈Θ_θψ(_i, _0, ,)ψ(_j, _0, ,) w()d)^2b(_i,)]
    ≤   ∫_∈Θ_θ[(ψ(_i, _0, ,)ψ(_j, _0, ,))^2b(_i,)] w()d
    ≤   ∫_∈Θ_θλ_max(Σ_ψ)[ψ(_i, _0, ,)^2b(_i,)]w()d
    ≤   ∫_∈Θ_θλ_max(Σ_ψ())(Σ_ψ,b()) w()d.

                By the definition of K_i, we have
                
    [K_i^2b(_i,)]
    ≤   ∫_∈Θ_θ{[ J K()ψ(_i, _0, ,)ψ(_i, _0, ,) K()Jb(_i,)]} w()d
    ≤   ∫_∈Θ_θλ_max(Σ_ψ,b())(J K() K()J) w()d
    
                        =   λ_max^2(J)∫_∈Θ_θλ_max(Σ_ψ,b())(Σ_K()) w()d,

                and consequently,
                
    [(ψ_1(_i, _0) K_j)^2b(_i,)]
    
                        =   [K_jΣ_ψ_1,bK_j] 
    ≤   λ_max(Σ_ψ_1,b)λ_max^2(J)∫_∈Θ_θλ_max(Σ_ψ())(Σ_K()) w()d,

                and
                
    [(ψ_1(_j, _0) K_i)^2b(_i,)]
    
                        =   [K_iΣ_ψ_1K_ib(_i,)] 
    ≤   λ_max(Σ_ψ_1)λ_max^2(J)∫_∈Θ_θλ_max(Σ_ψ,b())(Σ_K()) w()d.

                And similarly, it can be obtained that
                
    [(ψ_1(_i, _0) Hψ_1(_j, _0))^2b(_i,)]
                        ≤   λ_max(Σ_ψ_1,b)λ_max(Σ_ψ_1)(H^2).

                From the foregoing considerations, it is seen that by Assumption (A7),
                
    [h(_i,_j)^2b(_i,)]
    ≤   4[_ij(_0)b(_i,)]
       +4[(ψ_1(_i, _0) Hψ_1(_j, _0))^2 b(_i,)]
       +4[(ψ_1(_i, _0) K_j)^2 b(_i,)]+4[(ψ_1(_j, _0) K_i)^2 b(_i,)]
    ≤   4∫_∈Θ_θλ_max(Σ_ψ())(Σ_ψ,b()) w()d
       +4λ_max(Σ_ψ_1,b)λ_max^2(J)∫_∈Θ_θλ_max(Σ_ψ())(Σ_K()) w()d
       +4λ_max(Σ_ψ_1)λ_max^2(J)∫_∈Θ_θλ_max(Σ_ψ,b())(Σ_K()) w()d
       +4λ_max(Σ_ψ_1,b)λ_max(Σ_ψ_1)(H^2).

                
                Furthermore, we have
                
    [_ij(_0)b(_i,)b(_i,)b(_j,)]
    
                        =   [(∫_∈Θ_θψ(_i, _0, ,)ψ(_j, _0, ,) w()d)^2b(_i,)b(_j,)]
    ≤   ∫_∈Θ_θ[(ψ(_i, _0, ,)ψ(_j, _0, ,))^2b(_i,)b(_j,)] w()d
    ≤   ∫_∈Θ_θλ_max(Σ_ψ,b())[ψ(_i, _0, ,)^2b(_i,)]w()d
    ≤   ∫_∈Θ_θλ_max(Σ_ψ,b())(Σ_ψ,b())w()d,

                and
                
    [(ψ_1(_i, _0) K_j)^2b(_i,)b(_j,)]
    
                        =   [K_jΣ_ψ_1,bK_jb(_j,)] 
    ≤   λ_max(Σ_ψ_1,b)λ_max^2(J)∫_∈Θ_θλ_max(Σ_ψ,b())(Σ_K()) w()d,

                and
                
    [(ψ_1(_i, _0) Hψ_1(_j, _0))^2b(_i,)b(_j,)]
                        ≤   λ_max^2(Σ_ψ_1,b)(H^2).

                
                Therefore, we have
                
    [   h(_i,_j)^2b(_i,)b(_j,)]
    ≤   4[_ij(_0)b(_i,)b(_i,)b(_j,)]
       +4[(ψ_1(_i, _0) Hψ_1(_j, _0))^2 b(_i,)b(_j,)]
       +4[(ψ_1(_i, _0) K_j)^2 b(_i,)b(_j,)]+4[(ψ_1(_j, _0) K_i)^2 b(_i,)b(_j,)]
    ≤   4∫_∈Θ_θλ_max(Σ_ψ,b())(Σ_ψ,b())w()d
       +8λ_max(Σ_ψ_1,b)λ_max^2(J)∫_∈Θ_θλ_max(Σ_ψ,b())(Σ_K()) w()d
       +4λ_max^2(Σ_ψ_1,b)(H^2),

                which combining with (<ref>) and Lemmas <ref> and <ref> implies that
                
    [h(_1,_2)^2(1+b(_i,))(1+b(_j,))]

                is bounded, and consequently completes the proof. □
 
                
                    If Assumptions (A4) – (A5) and (A7) hold, we have for any fixed _1∈Θ_α,
                    
    [h(_1,_2)^2(1+b_1(_1,_1))(1+b_1(_2,_1))]

                    is bounded,
                    where b_1(,_1) is defined in Assumption (A7), h(_1,_2) is defined in (11) in the main paper, and _1 and _2 are generated from the null distribution with density f(,_0,,_0) and are independent of each other.	
                
                Proof of Lemma <ref>.	We defined Σ_ψ,b_1 and Σ_ψ_1,b_1 as following,
                
    Σ_ψ,b_1=   [b_1(_i,_1)ψ(_i, _0, ,)^⊗2],
    Σ_ψ_1,b_1=   [b_1(_i,_1)ψ_1(_i, _0)^⊗2].

                It is straightforward to complete the proof by the same arguments in Lemma <ref>. □
     
                Proof of Theorem 1.
For easy to express, we give some additional notations as follows.

    K =   ∫_∈Θ_θ K()w() d,
    _i =   ∫_∈Θ_θψ(_i, _0, , ) w()d,
    _ij() =   ∫_∈Θ_θψ(_i, , ,)ψ(_j, , ,) w()d.

By Lemma <ref>, we have

    ψ(, , ,)
    	=   ψ(, _0, ,)	- K()JΨ_1n(_0)+o_p(n^-1/2),

and

    1/n∑_i≠ j_ij()
    	=   1/n∑_i≠ j∫_∈Θ_θψ(_i, , ,)ψ(_j, , ,) w()d
    
    	=   1/n∑_i≠ j[_ij(_0) - 1/nΨ_1n(_0) K_i- 1/nK_jΨ_1n(_0) + 1/n^2Ψ_1n(_0) HΨ_1n(_0)]
       +1/n∑_i≠ j[_i	- 1/nKJΨ_1n(_0)] o_p(n^-1/2)
       +1/n∑_i≠ j[_j-1/nKJΨ_1n(_0)] o_p(n^-1/2)+o_p(n^-1).

By notations, we have, for any i≠ j,

    [K_i] =     [_ij(_0)] = 0.

It can be shown by the central limit theorem (CLT) that

    1/n∑_i≠ j[_i	- 1/nKJΨ_1n(_0)]
    	=   n-1/n∑_i=1^n[_i	- 1/nKJΨ_1n(_0)]
    
    	=   O_p(n^1/2),

which implies that

    T_n =   1/n(n-1)∑_i≠ j_ij()
    
    	=   1/n(n-1)∑_i≠ j[_ij(_0) - 1/nΨ_1n(_0) K_i- 1/nK_jΨ_1n(_0) + 1/n^2Ψ_1n(_0) HΨ_1n(_0)]
       +o_p(n^-1)
    ≡   T_n1+T_n2+T_n3+o_p(n^-1),

where

    T_n1 =   1/n(n-1)∑_i≠ j_ij(_0),
    
    	T_n2 =    -2/n^2∑_i=1^n Ψ_1n(_0) K_i,
    
    	T_n3 =   1/n^2Ψ_1n(_0) HΨ_1n(_0).

By some tedious calculations we have that

    T_n2
    	= -2/n^2∑_i=1^n[ψ_1(_i, _0) K_i]
    	= -2/n[ψ_1(_1, _0) K_1],

and

    T_n3
    	=1/n^2∑_k=1^n[ψ_1(_k, _0) Hψ_1(_k, _0)]
    	=1/n[ψ_1(_1, _0) Hψ_1(_1, _0)],

which implies that

    [T_n1+T_n2+T_n3]
    	=   -2/n[ψ_1(_1, _0) K_1]
    	+1/n[ψ_1(_1, _0) Hψ_1(_1, _0)]
       +O(n^-2)
    ≡   μ_0/n + O(n^-2).



Thus, we can rewrite T_n as

    T_n - μ_0/n = _n + R_n +o_p(n^-1),

where

    _n = 1/n(n-1)∑_i≠ jh(_i,_j)

with the kernel given in (11) in the main paper

    h(_i,_j)
        =   _ij(_0) - ψ_1(_i, _0) K_j
       - K_iψ_1(_j, _0) + ψ_1(_i, _0) Hψ_1(_j, _0)

and

    R_n =    -μ_0/n - 2/n^2∑_i=1^nψ_1(_i, _0) K_i
    	+1/n^2∑_i=1^nψ_1(_i, _0) Hψ_1(_i, _0)
       +2/n^2(n-1)∑_i≠ j^nψ_1(_i, _0) K_j
    	-1/n^2(n-1)∑_i≠ j^nψ_1(_i, _0) Hψ_1(_j, _0).

We are now going to verify that R_n=o_p(n^-1). In fact, we have, by the law of large numbers,

    -2/n^2∑_i=1^nψ_1(_i, _0) K_i
    	+1/n^2∑_i=1^nψ_1(_i, _0) Hψ_1(_i, _0)-μ_0/n = O_p(n^-3/2),

and

    2/n^2(n-1)∑_i≠ j^nψ_1(_i, _0) K_j
    	-1/n^2(n-1)∑_i≠ j^nψ_1(_i, _0) Hψ_1(_j, _0)
    	=o_p(n^-1).


Therefore, to complete the proof, it remains to show that

    n_n ζ,

where ζ is a random variable of the form ζ=∑_j=1^∞λ_j(χ^2_1j-1), and χ^2_11,χ^2_12,... are independent χ^2_1 variables, that is, ζ has characteristic function

    [e^itζ]=∏_j=1^∞(1-2itλ_j)^-1/2e^-itλ_j.

In fact, this can be shown by Theorem in 5.5.2 of <cit.> by checking its conditions.
It can be shown that

    g_1(v_1) = [h(_1,_2)|_1=v_1]=0,

and by Lemma <ref>,

    (h(_1,_2)) = [h(_1,_2)^2]=ν^2<∞.
 □

Proof of Theorem 2.

For easy to express, we use the same notations K, _i, K_i, H and _ij() in the proof of Theorem 1.
By Lemma <ref>, we have

    ψ(, , ,)
    	=   ψ(, _0, ,)	- K()JΨ_1n(_0)+o_p(n^-1/2),

and

    1/n∑_i≠ j_ij()
    	=   1/n∑_i≠ j∫_∈Θ_θψ(_i, , ,)ψ(_j, , , ) w()d
    
    	=   1/n∑_i≠ j[_ij(_0) - 1/nΨ_1n(_0) K_i- 1/nK_jΨ_1n(_0) + 1/n^2Ψ_1n(_0) HΨ_1n(_0)]
       +1/n∑_i≠ j[_i	- 1/nKJΨ_1n(_0)] o_p(n^-1/2)
       +1/n∑_i≠ j[_j-1/nKJΨ_1n(_0)] o_p(n^-1/2)+o_p(n^-1).

Under the alternative hypothesis and by Assumption (A2), we have, for any i≠ j,

    [_i]
    	=   ∫_∈Θ_θ[ψ(_i, _0, ,)] w()d
    
    	=   ∫_∈Θ_θ[ψ(_i, _0, , )] w()d
       - ∫_∈Θ_θ∂[ ψ(_i, _0, , )/∂] w()d(1+o(1))
    
    	=   O(n^-1/2),

and

    [ψ_1(_i,_0)]
    	= ∂[ψ_1(_i,_0)]/∂(1+o(1))
    	=O(n^-1/2).

It can be shown by the law of large numbers that

    1/n∑_i≠ j[_i	- 1/nKJΨ_1n(_0)]
    	=n-1/n∑_i=1^n[_i	- 1/nKJΨ_1n(_0)]
    	=O_p(n^1/2),

which implies that

    T_n =   1/n(n-1)∑_i≠ j_ij()
    
    	=   1/n(n-1)∑_i≠ j[_ij(_0) - 1/nΨ_1n(_0) K_i- 1/nK_jΨ_1n(_0) + 1/n^2Ψ_1n(_0) HΨ_1n(_0)]
       +o_p(n^-1)
    ≡   T_n1+T_n2+T_n3+o_p(n^-1),

where

    T_n1 =   1/n(n-1)∑_i≠ j_ij(_0),
    
    	T_n2 =    -2/n^2∑_i=1^n Ψ_1n(_0) K_i,
    
    	T_n3 =   1/n^2Ψ_1n(_0) HΨ_1n(_0).

By some tedious calculations we have that

    T_n2
    	= -2/n^2∑_i=1^n[ψ_1(_i, _0) K_i]
    	= -2/n[ψ_1(_1, _0) K_1],

and

    T_n3
    	=1/n^2∑_k=1^n[ψ_1(_k, _0) Hψ_1(_k, _0)]
    	=1/n[ψ_1(_1, _0) Hψ_1(_1, _0)],

which implies that

    [T_n2+T_n3]
    	=   -2/n[ψ_1(_1, _0) K_1]
    	+1/n[ψ_1(_1, _0) Hψ_1(_1, _0)]+O(n^-2)
    ≡   μ_0/n + O(n^-2).



Thus, we can rewrite T_n as

    T_n - μ_0/n = _n + R_n +o_p(n^-1),

where

    _n = 1/n(n-1)∑_i≠ jh(_i,_j)

with the kernel defined in (11) in the main paper

    h(_i,_j) = _ij(_0) - ψ_1(_i, _0) K_j- K_iψ_1(_j, _0) + ψ_1(_i, _0) Hψ_1(_j, _0)

and

    R_n =    -μ_0/n - 2/n^2∑_i=1^nψ_1(_i, _0) K_i
    	+1/n^2∑_i=1^nψ_1(_i, _0) Hψ_1(_i, _0)
       +2/n^2(n-1)∑_i≠ j^nψ_1(_i, _0) K_j
    	-1/n^2(n-1)∑_i≠ j^nψ_1(_i, _0) Hψ_1(_j, _0).

We are now going to verify that R_n=o_p(n^-1). In fact, we have, by the law of large numbers,

    -2/n^2∑_i=1^nψ_1(_i, _0) K_i
    	+1/n^2∑_i=1^nψ_1(_i, _0) Hψ_1(_i, _0)-μ_0/n = O_p(n^-3/2),

and

    2/n^2(n-1)∑_i≠ j^nψ_1(_i, _0) K_j
    	-1/n^2(n-1)∑_i≠ j^nψ_1(_i, _0) Hψ_1(_j, _0)
    	=o_p(n^-1).


Therefore, we complete the proof if we can show that

    n_n ζ_1,

and consequently by Theorem 9.5.2 of <cit.>, it suffices to show that for each t∈,

    |[e^it (n-1)_n]-[e^it ζ_1]|→ 0,

as n goes to infinity.

Let {_0i,i=1,⋯,n} be independent and identically distributed sample for each _0i from the null distribution with density f(,_0,,_0). We show this along the lines of Theorem in Section 5.5.2 of <cit.>. As shown in Theorem in Section 5.5.2 of <cit.>, we rewrite, by the representation of h(_1,_2) (Page 1087 in <cit.>),

    h(_1,_2) = ∑_k=1^∞λ_kϕ_k(_1)ϕ_k(_2),

where {ϕ_k(·)} and {λ_k} have following properties that

    [ϕ_j(_01)ϕ_k(_01)] = {[         1, if  j = k,;         0, otherwise, ].


    [ϕ_k(_01)]=0,    k,

and

    [h(_01,_02)^2]=∑_k=1^∞λ_k^2<∞.

Let

    T_nK = 1/n∑_i≠ j∑_k=1^Kλ_kϕ_k(_i)ϕ_k(_j).

According to the fact that |e^it-1|≤ |t|, we have

    |[e^it (n-1)_n]-[e^it T_nK]|
    	≤   [|e^it (n-1)_n-e^it T_nK|]
    ≤    |t|[|(n-1)_n-T_nK|]
    ≤   |t|[((n-1)_n-T_nK)^2]^1/2.

Next we claim that

    ((n-1)_n-T_nK)^2≤ 4(2C_ξ^2+1)∑_k=K+1^∞λ_k^2+(∑_k=K+1^∞λ_kμ_ak^2)^2.

In fact, (n-1)_n-T_nK has basically the form of a U-statistic, that is,

    (n-1)_n-T_nK=2/nn2U_nK,

where

    U_nK=1/n(n-1)∑_i≠ jg_K(_i,_j)

with

    g_K(_i,_j)
    	=   h(_i,_j)-∑_k=1^Kλ_kϕ_k(_i)ϕ_k(_j)
    
    	=   ∑_k=K+1^∞λ_kϕ_k(_i)ϕ_k(_j).


Before showing the inequality (<ref>), we derive following useful inequalities. 
It is seen that

    [ψ(_i,_0,  0, )]
    	=   [ψ(_0i,_0,  0, )f(_0i,_0,,_0)/f(_0i,_0,,_0)],

and consequently, by  Assumption (A6) and Cauchy-Schwarz inequality, there is a constant C_ such that

    ψ(_i,_0,  0, )≤1/√(n)C_(ψ(_0i,_0,  0, )^2)^1/2.

By Assumption (A5), it is followed from the above that there is a constant C_1>0 such that

    [_ij(_0)]
    	=   ∫_∈Θ_θ[ψ(_i,_0,  0, )]^2w() d
    ≤   1/nC_^2∫_∈Θ_θψ(_0i,_0,  0, )^2w() d
    
        =   1/nC_1C_^2,

and

    [K_i]
    	=   ∫_∈Θ_θJ K()[ψ(_i,_0,  0, )]w() d,

and by Assumption (A3), there is a constant C_3>0 such that


    [K_i]^2
    	≤   ∫_∈Θ_θJ K()[ψ(_i,_0,  0, )]^2w() d
    ≤   λ_max^2(J)∫_∈Θ_θλ_max(K() K())[ψ(_i,_0,  0, )]^2w() d
    ≤   1/nC_^2λ_max^2(J)∫_∈Θ_γλ_max(K() K())ψ(_0i,_0,  0, )^2w() d
    ≤   1/nC_3C_^2.

As the same arguments, we have

    [ψ_1(_i,_0)]
    	=[ψ_1(_0i,_0)f(_0i,_0,,_0)/f(_0i,_0,,_0)],

and there are constant C_2>0,

    [ψ(_i,_0)]≤   1/√(n)C_{[ψ_1(_0i,_0)^2]}^1/2
    ≤   1/√(n)C_2C_




From the foregoing considerations, we have

    |[h(_i,_j)]|
    		=   ∫_∈Θ_γ[ψ(_i,  0, )]^2w() d
       +[ψ_1(_i, _0) K_j+K_iψ_1(_j, _0)]
       + [ψ_1(_i, _0) Hψ_1(_j, _0)]
    ≤   1/nC_1C_^2+21/nC_3C_^2+1/nC_2C_^2
    
    		=   O(n^-1),

and by the mean value theorem and combing the dominated convergence theorem and Assumption (A6),

    [ϕ_k(_i)]
    	=   [ϕ_k(_0i)f(_0i,_0,,_0)/f(_0i,_0,,_0)]
    
    	=   1/n^1/2[ϕ_k(_0i)1/f(_0i,_0,,_0)∂ f(_0i,_0,,_0)/∂]
    
    	=   1/n^1/2[ϕ_k(_0i)1/f(_0i,_0,,_0)∂ f(_0i,_0,,_0)/∂](1+o(1))
    
    	=   1/n^1/2μ_ak+o(n^-1/2),

where  is between  and ,

    |[ϕ_k(_i)]|
    	≤   1/√(n)C_([ϕ_k(_0i)^2])^1/2
    
    	=   1/√(n)C_,

and

    [ϕ_k(_i)^2]
    	=   [ϕ_k(_0i)^2f(_0i,_0,,_0)/f(_0i,_0,,_0)]
    ≤   1+1/√(n)[ϕ_k(_0i)^2b(_0i,)]
    
    	=   1+O(n^-1/2).

Since by Lemma <ref> combining Assumption (A6),

    [g_K(_i,_j)^2]
    	=   ∫∫ g_K(_1,_2)^2f(_1,_0,,_0)f(_2,_0,,_0)d_1d_2
    ≤   [g_K(_0i,_0j)^2(1+n^-1/2b(_0i,))(1+n^-1/2b(_0j,))]
    ≤   [g_K(_0i,_0j)^2(1+n^-1/2b(_0i,))^2]
    
    	=   ∑_k=K+1^∞λ_k^2[ϕ_k(_0i)^2(1+n^-1/2b(_0i,))^2][ϕ_k(_0j)^2]
    
    	=   ∑_k=K+1^∞λ_k^2[ϕ_k(_0i)^2(1+n^-1/2b(_0i,))^2]
    ≤   (1+3Cn^-1/2)∑_k=K+1^∞λ_k^2
    
    	<   ∞,

where C is a constant such that [ϕ_k(_i)^2b(_0i,)^2]≤ C,
thus, it is straightforward to obtain that by the dominated theorem and for large n,

    [g_K(_i,_j)]
    	=   ∑_k=K+1^∞λ_k([ϕ_k(_i)])^2
    
    	=   1/n(∑_k=K+1^∞λ_kμ_ak^2)(1+o(1)),

and

    [g_K(_i,_j)^2]
    	=   ∫∫ g_K(_1,_2)^2f(_1,_0,,_0)f(_2,_0,,_0)d_1d_2
    ≤   [g_K(_0i,_0j)^2](1+o(1))
    
    	=   (1+o(1))∑_k=K+1^∞λ_k^2
    ≤   2∑_k=K+1^∞λ_k^2,

which implies that

    (g_K(_i,_j))
    	≤[g_K(_i,_j)^2]
    	≤2∑_k=K+1^∞λ_k^2.

Let b()=[g_K(_i,_j)|_j=]. Noting that

    b()=   [g_K(_i,_j)|_j=]
    
    	=   ∑_k=K+1^∞λ_kϕ_k()[ϕ_k(_i)],

we have, along same lines as (<ref>), for n large enough,

    [b(_i)^2]
    	=   [b(_0i)^2f(_0i,_0,,θ_0)/f(_0i,_0,,θ_0)]
    ≤   [b(_0i)^2(1+n^-1/2b(_0i,))]
    
    	=   [([g_K(_i,_j)|_i=_0i])^2(1+n^-1/2b(_0i,))]
    ≤   [[g_K(_i,_j)^2(1+n^-1/2b(_0i,))|_i=_0i]]
    
    	=   [g_K(_0i,_j)^2(1+n^-1/2b(_0i,))]
     =   [g_K(_0i,_0j)^2(1+n^-1/2b(_0i,))
          f(_0j,_0,,θ_0)/f(_0j,_0,,θ_0)]
    ≤   [g_K(_0i,_0j)^2(1+n^-1/2b(_0i,))(1+n^-1/2b(_0j,))]
    
    	<   ∞,

and consequently by the dominated theorem,

    ([g_K(_i,_j)|_i])
    	≤   [b(_i)^2]
    
    	=   [b(_0i)^2](1+o(1))
    
    	=   (1+o(1))∑_k=K+1^∞λ_k^2([ϕ_k(_j)])^2
    ≤   2/nC_ξ^2∑_k=K+1^∞λ_k^2.

Lemma A in Section 5.2.1 of <cit.> states that

    (U_nK)
    	≤   n2^-1(4C_ξ^2∑_k=K+1^∞λ_k^2+2∑_k=K+1^∞λ_k^2)
    ≤   2(2C_ξ^2+1)n2^-1∑_k=K+1^∞λ_k^2,

which implies that

    ((n-1)_n-T_nK)^2
    	=   (n-1)^2((U_nK)+([U_nK])^2)
    ≤   4(2C_ξ^2+1)∑_k=K+1^∞λ_k^2+(n-1)^2([U_nK])^2,

and consequently (<ref>) holds.


It is immediately obtained that n[h(_i,_j)]<+∞ by (<ref>), which implies that the series

    ∑_k=1^∞λ_kμ_ak^2=n[h(_i,_j)]<∞,

and consequently that

    lim_K→∞∑_k=K+1^∞λ_kμ_ak^2=0.

For fixed t and given >0, choose and fix K large enough such that

    2|t|(∑_k=K+1^∞λ_k^2)^1/2   <,   ,
    
    	|t|∑_k=K+1^∞λ_kμ_ak^2   <.

Then it can been shown that for all n sufficiently large,

    |t|^2([U_nK])^2=|t|^2(∑_k=K+1^∞λ_kμ_ak^2)^2<^2,

and immediately,

    |[e^it (n-1)_n]-[e^it T_nK]|<(2C_^2+2)^1/2.


We now are going to show that

    T_nK Y_K=∑_k=1^Kλ_k(W_k^2-1),

where {W_k,k=1,⋯,K} are independent and each W_k is normally distributed with mean μ_ak and unit variance.
We can rewrite

    T_nK= ∑_k=1^Kλ_k(W_nk^2-Z_nk),

where

    W_nk= 1/√(n)∑_i=1^nϕ_k(_i),

and

    Z_nk= 1/n∑_i=1^nϕ_k^2(_i).

Recalling the previous arguments, it can be obtained that

    [	W_nk]= 1/√(n)∑_i=1^n1/√(n)μ_ak(1+o(1))=μ_ak(1+o(1)),


    (W_nk)= 1/n∑_i=1^n(ϕ_k(_i))=1+O(n^-1/2)-1/nμ_ak^2→1,

and if j≠ k,

    (W_nj, W_nk)
    	=   [W_njW_nk] - [	W_nj][	W_nk]
    
    	=   1/n∑_i=1^n[ϕ_j(_i)ϕ_k(_i)]+1/n∑_i≠ l^n[ϕ_j(_i)ϕ_k(_l)]-μ_ajμ_ak(1+o(1))
    
    	=   1/n∑_i=1^n[ϕ_j(_i)ϕ_k(_i)]+1/n∑_i≠ l^n1/nμ_ajμ_ak(1+o(1))-μ_ajμ_ak(1+o(1))
    
    	=   [ϕ_j(_i)ϕ_k(_i)]-1/nμ_ajμ_ak(1+o(1))
    
    	=   [ϕ_j(_0i)ϕ_k(_0i)](1+n^-1/2C_)-1/nμ_ajμ_ak(1+o(1))
    →   0,

which implies that if j≠ k,

    |(W_nj, W_nk)|→0,

and consequently,

    (W_nj, W_nk)→{[         1, if  j = k,;         0, otherwise. ].

Therefore, by Lindeberg-Levy central limit theorem, we have

    (W_n1,⋯, W_nK) N(_a, _K× K),

where _a=(μ_a1,⋯,μ_aK). Since [ϕ_k^2(_i)]→1, it is seen that by the strong law of large numbers,

    Z_nk 1,

which implies that

    (Z_n1,⋯,Z_nK) (1,⋯,1).

Thus, (<ref>) holds and consequently,

    |[e^it T_nK]-[e^it Y_K]|<,

for all n sufficiently large.

Noting that by (<ref>) and (<ref>), for all n large enough,

    |[e^it (n-1)_n]-[e^it ζ_1]|
    	≤   |[e^it (n-1)_n]-[e^it T_nK]|
    	+|[e^it T_nK]-[e^it Y_K]|
       +|[e^it Y_K]-[e^it ζ_1]|
    
    	<   (2C_^2+2)^1/2+ |[e^it Y_K]-[e^it ζ_1]|.

Therefore, to complete the proof, it suffices to establish that for all K sufficiently large,

    |[e^it Y_K]-[e^it ζ_1]|< {[(W_1^2-1)^2]}^1/2.

Let the random variable W_1, W_2, ⋯ be defined on a common probability space and denote by ζ_1 the limit in mean square of Y_K as K→∞. Then, by the definition of Y_K, we have

    |[e^it Y_K]-[e^it ζ_1]|
    	≤    |t|[(Y_K-ζ_1)^2]^1/2
    ≤   |t|[(W_1^2-1)^2]^1/2(∑_k=K+1^∞λ_k^2)^1/2
    
    	<   [(W_1^2-1)^2]^1/2,

which arrives at (<ref>), and then completes the proof. □

Proof of Theorem 3.
It is straightforward to complete the proof if we have nT_n^*-μ_0 ζ, where ζ is defined in Theorem 1. Thus, it suffices to show that

    nT_n^*-μ_0 ζ.

Since the above asymptotic distribution is derived given the observed sample {_i, i=1,⋯,n}, we can treat  as a deterministic sequence satisfying  -_0=1/√(n)_1 for fixed _1∈Θ_α, where we use the fact that  is √(n)-consistent by Lemma <ref>.

Denote by ^* the expectation generated by the resampling from .
We give additional notations as follows.

    K^*()=   ∂^*ψ(^*, , , )/∂,
    
    	K^* =   ∫_∈Θ_γ K^*()w() d,
    
    	H^*=   ∫_∈Θ_γJ K^*() K^*()J w()d,
    _i^* =   ∫_∈Θ_γψ(_i^*, ^*, , ) w()d,
    _i^* =   ∫_∈Θ_γJ K^*() S(_i^*, ^*, ,) w()d,
    
    	K_i^* =   ∫_∈Θ_γJ K()ψ(_i^*, ^*, ,) w()d,
    _ij^*() =   ∫_∈Θ_γψ(_i^*, , ,)ψ(_j^*, , ,) w()d.


By Lemma <ref>, we have

    ψ(_i^*, ^*, ,)
    	=   ψ(_i^*, , ,)	- K^*()Jψ_1_i^*,)+o_p^*(n^-1/2),

and

    1/n∑_i≠ j_ij^*(^*)
    	=   1/n∑_i≠ j∫_∈Θ_γψ(_i^*, ^*, ,)ψ(_j^*, ^*, ,) w()d
    
    	=   1/n∑_i≠ j[_ij^*() - 1/nΨ_0n()_i^*- 1/n_j^*Ψ_1n()  + 1/n^2Ψ_1n() H^*Ψ_1n()]
       +1/n∑_i≠ j[_i^* - 1/n^*JΨ_1n()] o_p^*(n^-1/2)
       +1/n∑_i≠ j[_j^*-1/n^*JΨ_1n()] o_p^*(n^-1/2)+o_p^*(n^-1).

By above notations, we have, for any i≠ j,

    ^*[_i^*] =     ^*[_ij^*()] = 0.

It can be shown by the law of large numbers applying to bootstrap sample {_i^*,i=1, ⋯,n} that

    1/n∑_i≠ j[_i^*	- 1/n^*JΨ_1n()]
    	=n-1/n∑_i=1^n[_i^*	- 1/n^*JΨ_1n()]
    	=O_p^*(n^1/2),

which implies that

    T_n^* =   1/n(n-1)∑_i≠ j_ij^*(^*)
    
    	=   1/n(n-1)∑_i≠ j[_ij^*() - 1/nΨ_1n()_i^*- 1/n_j^*Ψ_1n() + 1/n^2Ψ_1n() H^*Ψ_1n()]
       +o_p(n^-1)
    ≡   T_n1^*+T_n2^*+T_n3^*+o_p^*(n^-1),

where

    T_n1^* =   1/n(n-1)∑_i≠ j_ij^*(),
    
    	T_n2^* =    -2/n^2∑_i=1^n Ψ_1n()_i^*,
    
    	T_n3^* =   1/n^2Ψ_1n() H^*Ψ_1n().

Furthermore, we have that

    ^* [T_n2^*]
    	= -2/n^2∑_i=1^n^*[ψ_1(^*, )_i^*]
    	= -2/n^*[ψ_1(, ) K_1^*],

and

    ^*[ T_n3^*]
    	=1/n^2∑_k=1^n^*[ψ_1(^*, ) H^*ψ_1(_k, )]
    	=1/n^*[ψ_1(^*, ) H^*ψ_1(^*, )],

which implies that

    ^*[T_n1^*+T_n2^*+T_n3^*]
    	=   -2/n^*[ψ_1(^*, )_1^*]
    	+1/n^*[ψ_1(^*, ) H^*ψ_1(^*, )]+O(n^-2)
    ≡   μ_0()/n + O(n^-2).

By the continuity of μ_0(), we have

    ^*[T_n1^*+T_n2^*+T_n3^*] = μ_0()/n(1+o_p(1)).


By the Assumption (A7) and the dominated convergence theorem, it is seen that for all ∈Θ_γ,

    K^*()
    	=   ∂∫ψ(, , , )f(,,,)d/∂
    
    	=   (1+O(n^-1/2))∂∫ψ(, _0, , )f(,_0,,)d/∂
    
    	=   K()(1+O(n^-1/2)),

and consequently,

    K^*=K(1+o(1)),
    
    	H^*=H(1+o(1)).

By the same arguments, we have

    μ_0() = μ_0(1+o(1)).

Thus, we can rewrite T_n^* as

    T_n^* - μ_0/n(1+o(1)) = _n^*(1+o_p^*(1)) + R_n^* +o_p^*(n^-1),

where

    _n^* = 1/n(n-1)∑_i≠ jh(_i^*,_j^*,),


    R_n^* =    -μ_0()/n - 2/n^2∑_i=1^nψ_1(_i^*, )_i^*
    	+1/n^2∑_i=1^nψ_1(_i^*, ) H^*ψ_1(_i^*, )
       +2/n^2(n-1)∑_i≠ j^nψ_1(_i^*, )_j^*
    	-1/n^2(n-1)∑_i≠ j^nψ_1(_i^*, ) H^*ψ_1(_j^*, ),

and

    h(_i^*,_j^*,)
    	=   _ij^*() - ψ_1(_i^*, ) K_j^*
       - (K_i^*)ψ_1(_j^*, ) + ψ_1(_i^*, ) H ψ_1(_j^*, ).


It can be verified that R_n^*=o_p^*(n^-1). In fact, we have, by the law of large numbers applying to bootstrap sample {_i^*,i=1, ⋯,n},

    -2/n^2∑_i=1^nψ_1(_i^*, )_i^*
    	+1/n^2∑_i=1^nψ_1(_i^*, ) H^*ψ_1(_i^*, )-μ_0()/n = O_p^*(n^-3/2),

and

    2/n^2(n-1)∑_i≠ j^nψ_1(_i^*, )_j^*
    	-1/n^2(n-1)∑_i≠ j^nψ_1(_i^*, ) H^*ψ_1(_j^*, )
    	=o_p^*(n^-1).


Next, we are going to show that

    n_n^* ζ,

where ζ is a random variable of the form ζ=∑_j=1^∞λ_j(χ^2_1j-1), and χ^2_11,χ^2_12,⋯ are independent χ^2_1 variables, that is, ζ has characteristic function

    [e^itζ]=∏_j=1^∞(1-2itλ_j)^-1/2e^-itλ_j.

Here {λ_j} are the eigenvalues of kernel h(_1,_2) under f(,_0,,_0), that is, they are the solutions of λ_jg_j(_2)=∫_0^∞h(_1,_2)g_j(_1)f(_1)d_1 for non-zero g_j.

Thus, by Theorem 9.5.2 of <cit.>, it remains to show that for each t∈,

    |[e^it (n-1)_n^*]-[e^it ζ]|→ 0,

as n goes to infinity.

Let {_0i,i=1,⋯,n} be independent and identically distributed sample for each _0i from the null distribution with density f(,_0,,_0), where {_0i,i=1,⋯,n} are independent of the observed sample {_i,i=1,⋯,n}. As shown in Theorem in Section 5.5.2 of <cit.>, we rewrite, by the representation of h(_1,_2) (Page 1087 in <cit.>),

    h(_1,_2) = ∑_k=1^∞λ_kϕ_k(_1)ϕ_k(_2),

where {ϕ_k(·)} and {λ_k} have following properties that

    [ϕ_j(_01)ϕ_k(_01)] = {[         1, if  j = k,;         0, otherwise, ].


    [ϕ_k(_01)]=0,    k,

and

    [h(_01,_02)^2]=∑_k=1^∞λ_k^2<∞.


Let

    T_nK = 1/n∑_i≠ j∑_k=1^Kλ_k(ϕ_k(_i^*)-[ϕ_k(_i^*)])(ϕ_k(_j^*)-[ϕ_k(_j^*)]).

First, we are going to show that for any fixed K,

    T_nK Y_K=∑_k=1^Kλ_k(W_k^2-1),

where {W_k,k=1,⋯,K} are independent and each W_k is normally distributed with mean μ_ak and variance 1.
We can rewrite

    T_nK= ∑_k=1^Kλ_k(W_nk^2-Z_nk),

where

    W_nk= 1/√(n)∑_i=1^n(ϕ_k(_i^*)-[ϕ_k(_i^*)]),


    Z_nk= 1/n∑_i=1^n(ϕ_k(_i^*)-[ϕ_k(_i^*)])^2.

By the mean value theorem and combing the dominated convergence theorem and Assumption (A7), it is seen that

    [ϕ_k(_i^*)]
    	=   [ϕ_k(_0i)f(_0i,,,_0)/f(_0i,_0,,_0)]
    	=   1/n^1/2[ϕ_k(_0i)_1∂ f(_0i,',,_0)/∂/f(_0i,_0,,_0)]
    
    =   1/n^1/2[ϕ_k(_0i)_1∂ f(_0i,_0,,_0)/∂/f(_0i,_0,,_0)](1+o(1))
    
    	=   1/n^1/2μ_ak+o(n^-1/2),

where ' is between  and , and then, we have

    |[ϕ_k(_i^*)]|
    	≤1/√(n)C_f(_1),

and

    [ϕ_k(_i^*)^2]
    	=   [ϕ_k(_0i)^2f(_0i,,,_0)/f(_0i,_0,,_0)]
    ≤   1+1/√(n)[ϕ_k(_0i)^2b_1(_0i,_1)]
    
    	=   1+1/√(n)C_f(_1).

Thus, it can be obtained that

    [W_nk]= 0,


    (W_nk)= 1/n∑_i=1^n(ϕ_k(_i^*))=1+O(n^-1/2)-1/nμ_ak^2→1,

and if j≠ k, by assumption (A7),

    (W_nj, W_nk)
    	=   [W_njW_nk] - [	W_nj][	W_nk]
    
    	=   1/n∑_i=1^n{[ϕ_j(_i^*)ϕ_k(_i^*)]-[ϕ_j(_i^*)][ϕ_k(_i^*)]}
    
    	=   [ϕ_j(_i^*)ϕ_k(_i^*)]-[ϕ_j(_i^*)][ϕ_k(_i^*)]
     =   1/n^1/2[ϕ_j(_0i)ϕ_k(_0i)_1∂ f(_0i,,,_0)/∂/f(_0i,_0,,_0)]
       -[ϕ_j(_i^*)][ϕ_k(_i^*)]
    ≤   1/n^1/2[|ϕ_j(_0i)ϕ_k(_0i)|b_1(_0i,_1)]+1/nC_f(_1)
    
    	=   O(n^-1/2)+O(n^-1)
    →   0,

which implies that if j≠ k,

    |(W_nj, W_nk)|→0,

and consequently,

    (W_nj, W_nk)→{[         1, if  j = k,;         0, otherwise. ].

Therefore, by Lindeberg-Levy central limit theorem, we have

    (W_n1,⋯, W_nK) N(_K, _K× K).

Since [ϕ_k^2(_i^*)]→1 and ([ϕ_k(_i^*)])^2→0, it is seen that by the strong law of large numbers,

    Z_nk 1,

which implies that

    (Z_n1,⋯,Z_nK) (1,⋯,1).

Thus, (<ref>) holds and consequently,

    |[e^it T_nK]-[e^it Y_K]|<,

for all n sufficiently large.

If we have (we will show this later) that for fixed t and given >0, choose and fix K large enough such that for all n sufficiently large,

    |[e^it (n-1)_n^*]-[e^it T_nK]|<6(C_f(_1)+1),

which combing with (<ref>) implies that, for all n large enough,

    |[e^it (n-1)_n^*]-[e^it ζ]|
    	≤   |[e^it (n-1)_n^*]-[e^it T_nK]|
    	+|[e^it T_nK]-[e^it Y_K]|
       +|[e^it Y_K]-[e^it ζ]|
    
    	<   (2C_f^2(_1)+2)^1/2+ |[e^it Y_K]-[e^it ζ]|.

Therefore, to obtain the inequality (<ref>), it suffices to establish that for all K sufficiently large,

    |[e^it Y_K]-[e^it ζ]|< {[(W_1^2-1)^2]}^1/2.

Let the random variable W_1, W_2, ⋯ be defined on a common probability space and denote by ζ the limit in mean square of Y_K as K→∞. Then according to the fact that |e^it-1|≤ |t|, we have by the definition of Y_K,

    |[e^it Y_K]-[e^it ζ]|
    	≤    |t|[(Y_K-ζ)^2]^1/2
    ≤   |t|[(W_1^2-1)^2]^1/2[∑_k=K+1^∞λ_k^2]^1/2
    
    	<   [(W_1^2-1)^2]^1/2,

which arrives at (<ref>). Therefore, the inequality (<ref>) holds.

The proof will be completed if we claim that (<ref>) holds.
To facilitate the expression, we give some additional notations below. Let

    _nK = 1/n∑_i≠ j∑_k=1^Kλ_kϕ_k(_i^*)ϕ_k(_j^*),


    R_1nK = (n-1)∑_k=1^Kλ_k([ϕ_k(_i^*)])^2,


    R_2nK = n-1/n∑_i=1^n∑_k=1^Kλ_k[ϕ_k(_j^*)]ϕ_k(_i^*),


    T_0n^* = 1/n∑_i≠ jh(_i^*,_j^*),

and

    _0n^* = 1/n∑_i≠ j[h(_i^*,_j^*)|_i^*).

Thus, (n-1)_n^*-T_nK can be decomposed as follows,

    (n-1)_n^*-T_nK
    	=   (n-1)_n^*-T_0n^*+2_0n^*-[T_0n^*]
       +T_0n^*-_nK
       +[T_0n^*]-R_1nK
       +2R_2nK-2_0n^*
    ≡   Π_1+Π_2+Π_3+Π_4,

where

    Π_1=   1/n∑_i≠ j{h(_i^*,_j^*,)-h(_i^*,_j^*)
       +[h(_i^*,_j^*)|_j^*]+[h(_i^*,_j^*)|_i^*]-[h(_i^*,_j^*)]},

and

    Π_2=   T_0n^*-_nK,
    Π_3=   [T_0n^*]-R_1nK,
    Π_4=   2R_2nK-2_0n^*.

We will deal with Π_j one by one as follows, j=1,2,3,4.

Before showing the upper bounds, we derive following useful inequalities. By the mean value theorem, it is seen that

    [ψ(_i^*,_0,  0, )]
    	=[ψ(_0i,_0,  0, )f(_0i,,,_0)/f(_0i,_0,,_0)],

and consequently, by  Assumption (A7) and Cauchy-Schwarz inequality, there are constant C_1>0 such that

    [ψ(_i^*,_0,  0, )]≤   C_f(_1)/√(n){[ψ(_0i,_0,  0, )^2]}^1/2
    ≤   1/√(n)C_1C_f(_1).

And it is followed from above inequality that

    [_ij^*(_0)]
    	=   ∫_∈Θ_θ[ψ(_i^*,_0,  0, )]^2w() d
    ≤   1/nC_1^2C_f^2(_1),

and

    [K_i^*]
    	=   ∫_∈Θ_θJ K()[ψ(_i^*,_0,  0, )]w() d,

and by Assumption (A3), there is a constant C_3>0 such that

    [K_i^*]^2
    	≤   ∫_∈Θ_θJ K()[ψ(_i^*,_0,  0, )]^2w() d
    ≤   ∫_∈Θ_θλ_max((K()JJ K()))[ψ(_i^*,_0,  0, )]^2w() d
    ≤   1/nC_1^2C_f^2(_1)∫_∈Θ_θλ_max((K()JJ K()))w() d
    ≤   1/nC_1^2C_3^2C_f^2(_1).

As the same arguments, we have

    [ψ_1(_i^*,_0)]
    	=[ψ_1(_0i,_0)f(_0i,,,_0)/f(_0i,_0,,_0)],

and there are constant C_2>0,

    [ψ_1(_i^*,_0)]≤   C_f(_1)/√(n){[ψ_1(_0i,_0)^2]}^1/2
    ≤   1/√(n)C_2C_f(_1).

And consequently, we have

    |[h(_i^*,_j^*)]|
    		=   |∫_∈Θ_θ[ψ(_i^*, _0, 0, )]^2w() d
       +[ψ_1(_i^*, _0) K_j^*+ ψ_1(_j^*, _0) K_i^*]
       + [ψ_1(_i^*, _0) Hψ_1(_j^*, _0)]|
    ≤   1/n[C_1^2+21/√(n)C_1C_2C_3+C_2^2λ_max(H)]C_f^2(_1)
    
    		=   O(n^-1).


To bound Π_1. Rewrite

    Π_1=(n-1)U_1n,

where

    U_1n=1/n(n-1)∑_i≠ jȟ(_i^*,_j^*),

and

    ȟ(_i^*,_j^*)
    	=   h(_i^*,_j^*,)-h(_i^*,_j^*)
       +[h(_i^*,_j^*)|_j^*)+[h(_i^*,_j^*)|_i^*)-[h(_i^*,_j^*)].

Since U_1n is a U-statistic with  [ȟ(_i^*,_j^*)|_j^*]=0, we have, by Lemma A in Section 5.2.1 of <cit.>,

    [Π_1^2]
    	=   (n-1)^2(U_1n)
    
    	=   (n-1)^2n2^-1(ȟ(_i^*,_j^*))
    ≤   2(ȟ(_i^*,_j^*))
    
    	=   2[ȟ(_i^*,_j^*)^2]
    ≤   4[(h(_i^*,_j^*,)-h(_i^*,_j^*)+[h(_i^*,_j^*)])^2]
       +16[([h(_i^*,_j^*)|_i^*)-[h(_i^*,_j^*)])^2].

According to Assumption (A7) and the mean value theorem, it is seen that for n large enough,

    [|h(_i^*,_j^*,)-h(_i^*,_j^*)|^2]
     =   [|h(_0i,_0j,)-h(_0i,_0j)|^2f(_0i,,,_0)/f(_0i,_0,,_0)f(_0j,,,_0)/f(_0j,_0,,_0)]
    ≤   1/n[(_1∂ h(_0i,_0j,)/∂)^2(1+b_1(_0i,_1))(1+b_1(_0j,_1))],

where  is between  and . By utilizing Assumption (A7), there is a constant C__1>0 such that

    [(_1∂ h(_0i,_0j,)/∂)^2]≤    C__1,
    [(_1∂ h(_0i,_0j,)/∂)^2b_1(_0i,_1)]≤    C__1,
    [(_1∂ h(_0i,_0j,)/∂)^2b_1(_0i,_1)b_1(_0j,_1)]≤    C__1,

which implies that

    [|h(_i^*,_j^*,)-h(_i^*,_j^*)|^2]=O(n^-1).



By the foregoing arguments, we have

    [h(_i^*,_j^*)|_i^*]
    	=   ∫_∈Θ_θ[ψ(_j^*,_0,  0, )]ψ(_i^*,_0,  0, ) w() d
       +ψ_1(_i^*, _0)[K_j^*]+ [ψ_1(_j^*, _0)] K_i^*
       + ψ_1(_i^*, _0) H[ψ_1(_j^*, _0)],

and consequently, for n large enough,

    [([h(_i^*,_j^*)|_i^*)])^2]
    ≤   3∫_∈Θ_θ[{[ψ(_j^*,_0,  0, )]ψ(_i^*,_0,  0, )}^2] w() d
       +6[{[ψ_1(_i^*, _0)][K_j^*]}^2]
       + 3[{[ψ_1(_i^*, _0)] H[ψ_1(_j^*, _0)]}^2]
    ≤   3∫_∈Θ_θ[ψ(_j^*,_0,  0, )]^2 [ψ(_i^*,_0,  0, )^2] w() d
       +6[ψ_1(_i^*, _0)^2][K_j^*]^2
       + 3λ_max(H)^2[ψ_1(_i^*, _0)^2][ψ_1(_j^*, _0)]^2
    ≤   3C_1^2C_f^2(_1)1/n(1+1/√(n)C_f(_1)C_M)
       +6C_1^2C_3^2C_f^2(_1)1/nC_M+3C_2^2C_f^2(_1)1/nC_M
    
    	=   O(n^-1),

where C_M is a constant.

Combining the above arguments with inequality (<ref>), we have

    [Π_1^2]
    	≤   8[{h(_i^*,_j^*,)-h(_i^*,_j^*)}^2]+8([h(_i^*,_j^*)])^2
       +32[{[h(_i^*,_j^*)|_i^*)}^2]+32([h(_i^*,_j^*)])^2
    
    	=   O(n^-1).




To bound Π_2.
Noting that T_0n^*-_nK has basically the form of a U-statistic, that is,

    T_0n^*-_nK=2/nn2U_nK,

where

    U_nK=1/n(n-1)∑_i≠ jg_K(_i^*,_j^*)

with

    g_K(_i^*,_j^*)
    	=   h(_i^*,_j^*)-∑_k=1^Kλ_kϕ_k(_i^*)ϕ_k(_j^*)
    
    	=   ∑_k=K+1^∞λ_kϕ_k(_i^*)ϕ_k(_j^*).

Next we claim that

    (T_0n^*-_nK)^2≤ 4(2C_f^2(_1)+1)∑_k=K+1^∞λ_k^2.


From inequality (<ref>) and (<ref>), it is straightforward to obtain that for large n,

    [g_K(_i^*,_j^*)]
    	=   ∑_k=K+1^∞λ_k([ϕ_k(_i^*)])^2
    
    	=   1/n(∑_k=K+1^∞λ_k_ak^2)(1+o(1)),

where

    _ak=[ϕ_k(_0i)_2∂ h(_1,_2, )/∂].

Since by Lemma <ref> combining with Assumption (A7) and Cauchy-Schwarz inequality,

    [g_K(_i^*,_j^*)^2]
    
    	=   ∫∫ g_K(_1,_2)^2f(_1,,,_0)f(_2,,,_0)_1d_2
    ≤   [g_K(_0i,_0j)^2(1+n^-1/2b_1(_0i,_1))(1+n^-1/2b_1(_0j,_1))]
    ≤   [g_K(_0i,_0j)^2(1+n^-1/2b_1(_0i,_1))^2]
    
    	=   ∑_k=K+1^∞λ_k^2[ϕ_k(_0i)^2(1+n^-1/2b_1(_0i,_1))^2][ϕ_k(_0j)^2]
    
    	=   ∑_k=K+1^∞λ_k^2[ϕ_k(_0i)^2(1+n^-1/2b_1(_0i,_1))^2]
    ≤   (1+3Cn^-1/2)∑_k=K+1^∞λ_k^2
    	
    	<   ∞,

where C is a constant such that [ϕ_k(_i)^2b_1(_0i,_1)^2]≤ C,
thus, it is straightforward to obtain that by the dominated theorem and for large n,

    [g_K(_i,_j)^2]
    	=   ∫∫ g_K(_1,_2)^2f(_1,,,_0)f(_2,,,_0)d_1d_2
    ≤   [g_K(_0i,_0j)^2](1+o(1))
    
    	=   (1+o(1))∑_k=K+1^∞λ_k^2
    ≤   2∑_k=K+1^∞λ_k^2,

which implies that

    (g_K(_i^*,_j^*))
    	≤[g_K(_i^*,_j^*)^2]
    	≤2∑_k=K+1^∞λ_k^2.

Let b()=[g_K(_i^*,_j^*)|_j^*=]. Noting that

    b()=   [g_K(_i^*,_j^*)|_j^*=]
    
    	=   ∑_k=K+1^∞λ_kϕ_k()[ϕ_k(_i^*)],

we have as similar to (<ref>), for n large enough,

    [b(_i^*)^2]
    	=   [b(_0i)^2f(_0i,,,_0)/f(_0i,_0,,_0)]
    ≤   [b(_0i)^2(1+n^-1/2b_1(_0i,_1))]
    
    	=   [([g_K(_i,_j)|_i=_0i])^2(1+n^-1/2b_1(_0i,_1))]
    ≤   [[g_K(_i,_j)^2(1+n^-1/2b_1(_0i,_1))|_i=_0i]]
    
    	=   [g_K(_0i,_j)^2(1+n^-1/2b_1(_0i,_1))]
    
    	=   [g_K(_0i,_0j)^2(1+n^-1/2b_1(_0i,_1))f(_0j,,,_0)/f(_0j,_0,,_0)]
    ≤   [g_K(_0i,_0j)^2(1+n^-1/2b_1(_0i,_1))(1+n^-1/2b_1(_0j,_1))]
    
    	<   ∞,

and consequently by the dominated theorem,

    ([g_K(_i^*,_j^*)|_i^*])
    	≤   [b(_i^*)^2]
    
    	=   [b(_0i)^2](1+o(1))
    
    	=   (1+o(1))∑_k=K+1^∞λ_k^2([ϕ_k(_j)])^2
    ≤   2/nC_f^2(_1)∑_k=K+1^∞λ_k^2.

Lemma A in Section 5.2.1 of <cit.> states that

    (U_nK)
    	≤   n2^-1(4C_f^2∑_k=K+1^∞λ_k^2+2∑_k=K+1^∞λ_k^2)
    ≤   2(2C_f^2(_1)+1)n2^-1∑_k=K+1^∞λ_k^2,

which implies that

    (_n-_nK)^2
    	=   (n-1)^2((U_nK)+([U_nK])^2)
    ≤   4(2C_f^2(_1)+1)∑_k=K+1^∞λ_k^2,

and consequently (<ref>) holds.

To bound Π_3. We now show that for any large n,

    [|Π_3|]=[|T_0n^*-R_1nK|]→0,

as K goes infinity.
It is immediately obtained that n[h(_i^*,_j^*)]<+∞ by (<ref>), which implies that the series

    (n-1)∑_k=1^∞λ_k([ϕ_k(_i^*)])^2=(n-1)[h(_i^*,_j^*)]<∞,

and consequently that

    lim_K→∞(n-1)∑_k=K+1^∞λ_k([ϕ_k(_i^*)])^2=0.

It follows immediately that for any large n,

    [|T_0n^*-R_1nK|]
    	=(n-1)∑_k=K+1^∞λ_k([ϕ_k(_i^*)])^2→0,

as K goes infinity.

To bound Π_4. By the definition of R_2nK and _0n^*, it is seen that

    1/n-1(_0n^*-R_2nK)
    	=1/n∑_i=1^ng_1K(_i^*),

where

    g_1K(_i^*)
    	=∑_k=K+1^∞λ_k[ϕ_k(_j^*)]ϕ_k(_i^*).

Invoking the properties of ϕ_k(·), we have, by the inequality (<ref>) and (<ref>),

    [g_1K(_i^*)^2]
    	=   ∑_k=K+1^∞λ_k^2([ϕ_k(_j^*)])^2[ϕ_k(_i^*)^2]
    ≤   C_f^2(_1)1/n∑_k=K+1^∞λ_k^2,

which implies that

    [(_0n^*-R_2nK)^2]
    	=   (n-1)^2/n^2∑_i=1^n(g_1K(_i^*)^2)+(n-1)^2/n([g_1K(_i^*)])^2
    ≤   C_f^2(_1)(n-1)^2/n^2∑_k=K+1^∞λ_k^2+(n-1)^2/n([g_1K(_i^*)])^2
    ≤   C_f^2(_1)∑_k=K+1^∞λ_k^2+n([g_1K(_i^*)])^2.



We have obtained the upper bounds for all Π_j, j=1,2,3,4. For fixed t and given >0, choose and fix K large enough such that

    |t|(∑_k=K+1^∞λ_k^2)^1/2   <,  
    
    	|t|∑_k=K+1^∞λ_kμ_ak^2   <.

According to (<ref>), it is seen that

    [g_1K(_i^*)]
    	=   ∑_k=K+1^∞λ_k([ϕ_k(_i^*)])^2
    
    	=   1/n∑_k=K+1^∞λ_kμ_ak^2+o(n^-1).

Thus, by (<ref>), (<ref>) and (<ref>), we have

    |t|[|Π_2|]≤   |t|([Π_2^2])^1/2≤2(2C_f^2(_1)+1)^1/2
    
    	|t|[|Π_3|]≤   
    
    	|t|[|Π_4|]≤   |t|([Π_4^2])^1/2≤ 2(C_f(_1)+1)

According to the fact that |e^it-1|≤ |t|, we have, by the above arguments and inequality (<ref>) and for all n sufficiently large,

    |[e^it (n-1)_n^*]-[e^it T_nK]|
    	≤   [|e^it (n-1)_n^*-e^it T_nK|]
    ≤    |t|[|(n-1)_n^*-T_nK|]
    ≤    |t|{[|Π_1|]+[|Π_2|]+[|Π_3|]+[|Π_4|]}
    
    	<    6(C_f(_1)+1),

which establishes (<ref>) and then completes the whole proof. □



§ SIMULATION STUDIES FOR LARGE 

B.table
B.figure



 §.§ Change plane analysis with sparse  for quantile, probit and semiparametric models


For fair comparison, in <ref>–<ref>, we consider the settings as similar to the existing literature except for large q <cit.>.

We consider the probit regression model in the main paper

    f(_i, , , )=Φ(0.5+_1i+_i(_i≥0))^Y_i+Φ(-0.5-_1i-_i(_i≥0))^1-Y_i.

We generate X_1i independently from binomial distribution b(1,0.5), _i from p-variate normal distribution with mean  and variance √(2)I, and Z_1i=1 and (Z_2i,⋯,Z_qi) from (q-1)-variate standard normal distribution.
We set =(1,⋯,1) under H_0. Under H_1, θ_2,⋯,θ_11 are generated randomly from uniform distribution U(1, 3) and θ_12,⋯,θ_q are zero, where θ_1 is chosen as the negative of the 0.65 percentile of Z_2θ_2+⋯+Z_qθ_q, which means that  divides the population into two groups with 0.35 and 0.65 observations, respectively.

Consider the quantile regression model in the main paper,

    Y_i = 0.5+_i+_i(τ)(_i(τ)≥0) + _i,

where _i is generated from standard normal distribution, _i and _i are generated as same as these in probit regression model. The error _i is generated from t_3 distribution.
The parameters  under H_0 and  under H_1 are same as these in probit model.


Consider the semiparametric model in the main paper,

    Y_i = γ(_i,_2) +_i A_i (_i≥0)+_i.

As the same settings provided by <cit.>, we consider two settings for the propensity score model π(_i,_1), denoted by P1 and P2:

	
  * π(_i,_1)=0.5;
	
  * π(_i,_1)=exp(0.5_2i+0.5_3i)/1+exp(0.5_2i+0.5_3i),

and two baseline mean functions for γ(_i,_2), denoted by B1 and B2:

	
  * γ(_i,_2) = 1+0.5_2i+_3i^2;
	
  * γ(_i,_2) = 1+sin(_2i+_3i).

We calculate the WAST and SST in the main paper by fitting a linear model for baseline function γ(_i,_2) and a logistic regression for  the propensity score model π(_i,_1). Therefore, two baseline models γ(_i,_2) are misspecified. We generate v_1i independently from binomial distribution b(1,0.5), v_2i independently from uniform distribution U(-1,1), and X_ji independently from uniform distribution U(-1,1) , where i=1,⋯,n and j=1,⋯,p. We set _i=(1, v_1i, v_2i) and generate _i as same as these in probit regression model.
The parameters  under H_0 and  under H_1 are same as these in probit model.


For all models with change plane analysis, we evaluated the power under a sequence of alternative models indexed by κ, that is H_1^κ: ^κ=κ^* with κ = i/10 for semiparametric model and κ=i/20 for others, i=1,⋯,10, where ^*=(1,⋯,1). We set sample size n=(200, 400, 600) for quantile, probit and semiparametric models, 1000 repetitions and 1000 bootstrap samples, and report in Figure <ref>-<ref> the performance for the WAST and SST. We calculate the SST over {^(k)=(θ^(k)_1,⋯,θ^(k)_q): k=1,⋯,K} with the number of threshold values K=1000. Let ^(k)_-1=^(k)_-1/^(k)_-1, where ^(k)_-1=(θ^(k)_2,⋯,θ^(k)_q) and ^(k)_-1=(θ̃^(k)_2,⋯,θ̃^(k)_q), and ^(k)_-1 is drawn independently form (r-1)-variate standard normal distribution.  For each θ_1^(k), k=1,⋯,K, we selected it by equal grid search in the range from the lower 10th percentile to upper 10th percentile of the data points of {θ^(k)_2Z_2i+⋯+θ^(k)_qZ_qi}_i=1^n, which is same as that in <cit.>. Here we consider four combinations of (p,q)=(2, 100), (2, 500), (6, 100), (6, 500), (11, 100), (11, 500).

Type 1 errors (κ=0) for probit, quantile and semiparametric models are listed in Table <ref>. We can see from Table <ref> that the size of the proposed WAST are close to the nominal significance level 0.05, but for most scenarios the size of the SST are much smaller than 0.05.
Figure <ref>-<ref> indicate that powers become greater as sample size n increases, which are verified the asymptotic theory. The proposed WAST has comparable power with the SST for the semiparametric model, but the size of the SST is much less than the nominal level 0.05.



















 §.§ Change plane analysis with dense  for quantile, probit and semiparametric models


For the sparse , we generate θ_2,⋯,θ_q from the uniform distribution U(0,3). Other settings are same as these in Section <ref>.

Type 1 errors (κ=0) for probit, quantile and semiparametric models are listed in Table <ref>. We can see from Table <ref> that the size of the proposed WAST are close to the nominal significance level 0.05, but for most scenarios the size of the SST are much smaller than 0.05. From Figure <ref>-<ref> we have same conclusion as these in Section <ref>. We omit the detailed analysis here.




















 §.§ Change plane analysis with sparse  for GLMs


Consider the GLM in the main paper with the canonical parameter

    μ =  + (≥0).

For Gaussian and Poisson families,
we set α_1 = 0.5,  and for binomial family α_1 is chosen so that the proportion of cases in the data set is 1/3 on average under the null hypothesis. θ_2,⋯,θ_q are generated randomly from uniform distribution U(1, 3), and θ_1 is chosen as the negative of the 0.65 percentile of Z_2θ_2+⋯+Z_qθ_q, which means that  divides the population into two groups with 0.35 and 0.65 observations, respectively. For the predictor, we generate (Z_1, Z_2,⋯,Z_q) from a multivariate normal distribution with mean  and covariance _q. We set _j=(v_j>0) with j=2,⋯,r, X_k = (v_k>0) with k=2,⋯,p, and _1=1, X_1=1, and Z_1=1.

For all models with change plane analysis, we evaluated the power under a sequence of alternative models indexed by κ, that is H_1^κ: ^κ=κ^* with κ = i/10 for semiparametric model and κ=i/20 for others, i=1,⋯,10, where ^*=(1,⋯,1). We set sample size n=(300, 600) for GLMs, 1000 repetitions and 1000 bootstrap samples, and report in Figure <ref>-<ref> the performance for the WAST, SST and SLRT. We calculate the SST and SLRT over {^(k)=(θ^(k)_1,⋯,θ^(k)_q): k=1,⋯,K} with the number of threshold values K=1000. Let ^(k)_-1=^(k)_-1/^(k)_-1, where ^(k)_-1=(θ^(k)_2,⋯,θ^(k)_q) and ^(k)_-1=(θ̃^(k)_2,⋯,θ̃^(k)_q), and ^(k)_-1 is drawn independently form (r-1)-variate standard normal distribution.  For each θ_1^(k), k=1,⋯,K, we selected it by equal grid search in the range from the lower 10th percentile to upper 10th percentile of the data points of {θ^(k)_2Z_2i+⋯+θ^(k)_qZ_qi}_i=1^n, which is same as that in <cit.>. Here we consider four combinations of (p,q)=(2, 100), (2, 500), (6, 100), (6, 500), (11, 100), (11, 500).

Type 1 errors (κ=0) for GLMs are list in Table <ref>. We can see from Table <ref> that the sizes of the proposed WAST and the SLRT are close to the nominal significance level 0.05, but for most scenarios the size of the SST are much smaller than 0.05.
Figure <ref>-<ref> indicate that powers become greater as sample size n increases, which are verified the asymptotic theory. The proposed WAST has comparable power with the SST for the semiparametric model, but the size of the SST is much less than the nominal level 0.05 in 
Figure <ref>-<ref> show that the powers of the proposed WAST are uniformly greater than the competitors SST and SLRT.












 §.§ Change plane analysis with dense  for GLMs


For the sparse , we generate θ_2,⋯,θ_q from the uniform distribution U(0,3) and set the rest being zero. Other settings are same as these in Section <ref>.

Type 1 errors (κ=0) for probit, quantile and semiparametric models are listed in Table <ref>. We can see from Table <ref> that the size of the proposed WAST are close to the nominal significance level 0.05, but for most scenarios the size of the SST are much smaller than 0.05. From Figure <ref>-<ref> we have same conclusion as these in Section <ref>. We omit the detailed analysis here.











 §.§ Comparison of the WAST 

In this section, we compare the WAST with close form (15) and with approximation (14). The simulation settings are same as that in Section 5 of main paper, in which we set (r, p, q)=(2, 11, 100) and apply sparse scenario with the number of nonzero parameters being 10. We set N=500, 1000, 5000, 10000 in the formula (14) of the main paper. Figure <ref> indicate that the WAST has slightly greater power than the WAST with approximation for each distribution and sample size. The number N of the generated zs is not sensitive for the WAST with approximation when N≥500.





§ SIMULATION STUDIES FOR SMALL 



C.table
C.figure
C.thm



 §.§ Single change plane for GLMs

For fair comparison, we use the same settings as considered by <cit.>.
Consider the GLMs with the canonical parameter

    μ =  + (≥0),

where = κ_p and α_2=⋯=α_r=log(1.4). For Gaussian and Poisson families,
we set α_1 = 0.5,  and for binomial family α_1 is chosen so that the proportion of cases in the data set is 1/3 on average under the null hypothesis. θ_2,⋯,θ_q are equally spaced numbers from -1 to 1, and θ_1 is chosen as the negative of the 0.65 percentile of Z_2θ_2+⋯+Z_qθ_q, which means that  divides the population into two groups with 0.35 and 0.65 observations, respectively. For the predictor, we generate (v_2, ⋯,v_max{r,p}, Z_2,⋯,Z_q) from a multivariate normal distribution with mean  and covariance Σ=(ρ̃_ij), where ρ̃_ij=1 if i=j and ρ̃_ij=ρ otherwise. Here, we consider both ρ=0 and ρ=0.3. We set _j=(v_j>0) with j=2,⋯,r, X_k = (v_k>0) with k=2,⋯,p, and _1=1, X_1=1, and Z_1=1.

We evaluate the power under a sequence of alternative models indexed by κ, i.e., H_1^κ:  = κ_p with κ equally spaced in the range (0,κ_max]. We set the sample size as n=(300, 600), and we set 1000 repetitions and 1000 bootstrap samples. For comparison, we consider three methods here: (i) the proposed WAST, (ii) the SST, and (iii) the approximated supremum of likelihood-ratio test statistic (SLRT) given by <cit.>. We calculate both SLRT and SST over {^(k)=(θ^(k)_1,⋯,θ^(k)_q): k=1,⋯,K}. Let ^(k)_-1=^(k)_-1/^(k)_-1, where ^(k)_-1=(θ^(k)_2,⋯,θ^(k)_q) and ^(k)_-1=(θ̃^(k)_2,⋯,θ̃^(k)_q), and ^(k)_-1 is drawn independently from the multivariate normal distribution 𝒩(_q-1,_q-1). For each θ_1^(k), k=1,⋯,K, we select it by an equal grid search in the range from the lower tenth percentile to the upper tenth percentile of the data points of {θ^(k)_2Z_i2+⋯+θ^(k)_qZ_iq}_i=1^n, which is the same as that in <cit.>. Here, we set K=1000.

Table <ref> lists the type-1 errors (κ=0) with the nominal significance level of 0.05. As can be seen, the sizes of the proposed method are close to 0.05. For the SST, its sizes are far from 0.05 in most of the scenarios, and it performs the worst of these three methods. For the SLRT, its sizes are close to 0.05 except for (r,p,q)=(2, 51,11) and (r,p,q)=(6, 51,11).



Figure <ref> - <ref> show the powers for all the scenarios considered in Table <ref>. As can be seen, the powers increase as the sample size n increases, which verifies the asymptotic theory as expected. For the Gaussian, binomial and Poisson families, the power of the proposed WAST increases faster than those of the SST and SLRT, and this superiority is particularly evident when n is small or p and q increase. In summary, the proposed test statistic shows very competitive performance.











 §.§ Single change plane for GLMs with  from 

Consider the GLM in model (1.1) with the same setting as Section 5 but the predictors. Let =(ς_1, ⋯,ς_d), where d=max(r,p)+q-2. We generate ς_1, ⋯,ς_d independently from t_3 distribution with degrees of freedom 3. We set (v_2, ⋯,v_max{r,p}, Z_2,⋯,Z_r)=Σ^1/2, where Σ=(ρ̃_ij) with ρ̃_ij=1 if i=j and ρ otherwise. We set _j=(v_j>0) with j=2,⋯,r,  X_k = (v_k>0) with k=2,⋯,p, and _1=1, X_1=1 and Z_1=1. In this case, the range of  is larger than that when  from multivariate standard normal distribution as used in Section 5.

The type 1 errors of the proposed WAST and the SST and SLRT are listed in Table <ref>. From Table <ref>, it can be found that the type 1 errors for the proposed WAST and the SLRT are close to the nominal significant level 0.05 when p is small, such as p=2,6. When p is large (p=51), the WAST still achieves the type 1 error close to 0.05, while the type 1 errors of the SLRT are much larger than 0.05. As Table 1 in the main paper, the type 1 errors are far from 0.05 in most cases.

The power curves of the WAST, SST and SLRT are depicted in Figures <ref>-<ref>. From Figures <ref>-<ref>, we see that the proposed method have higher power uniformly than those of the SLRT and SST, which illustrates the robustness of the proposed method and motivates researchers to apply the proposed method in practice.

We also generate ς_1, ⋯,ς_d independently from normal distribution with mean zero and standard deviation 5. Since the range of  is large, the performance of the proposed test statistic are similar to that when ς_1, ⋯,ς_d are independently from t_3 distribution with degrees of freedom 3. We list the sizes in Table <ref> and depict power curves in Figures <ref>-<ref> of the proposed test with two competitors SST and SLRT.
The same performance the proposed test statistic can be found from Table <ref> and Figures <ref>-<ref>.


























 §.§ Multiple change-plane analysis


It is straightforward to extend the proposed test procedure for models with a change plane to the case of multiple change planes, as considered by <cit.>. For instance, we consider the quantile regression model with multiple change planes:

    Q_Y_i(τ|_i, _i, _i) = _i(τ)+∑_t=1^T_ti_t(τ)(_ti_t(τ)≥0),

where (τ)=(α_1(τ), ⋯, α_r(τ))∈Θ_α⊆ℝ^r, _t(τ)=(β_t1(τ), ⋯, β_tp_t(τ))∈Θ_β_t⊆ℝ^p_t, and _t(τ)=(θ_t1(τ), ⋯, θ_tq_t(τ))∈Θ__t⊆ℝ^q_t are unknown parameters, and the number of change planes T is a preset integer, _i=(_1i, ⋯, _Ti) and _i=(_1i, ⋯, _Ti). We still denote (τ)=(_1(τ), ⋯, _T(τ)) and (τ)=(_1(τ), ⋯, _T(τ)),
and Θ_β=Θ_β_1×⋯×Θ_β_T and Θ_θ=Θ_θ_1×⋯×Θ_θ_T.

Testing whether _t(τ)= for all t∈{1,⋯,T} is of interest, i.e.,

    H_0: (τ)=  versus   H_1: (τ)≠.

Similar to the WAST in the main paper, we propose the integrated statistic

    T_n^(m) = 1/n(n-1)∑_i≠ jω̃_ij[(Y_i-_i(τ)≤0)-τ][(Y_j-_j(τ)≤0)-τ]

with

    ω̃_ij = ∑_t=1^T_ti_tj∫__t∈Θ_θ_t(_ti_t≥0) (_tj_t≥0)w_t(_t)d_t.

Taking the weight w() to be the standard multivariate normal density leads to, for i≠ j,

    ω̃_ij=1/4∑_t=1^T_ti_tj+1/2π∑_t=1^T_ti_tjarctan(ϱ_t, ij/√(1-ϱ_t, ij^2)),

where ϱ_t, ij = _ti_tj(_ti_tj)^-1.

As in model (7) in the main paper, we consider the empirical estimating functions

    Ψ_n(, , )=   ∑_i=1^n_i()ψ_0^(m)(_i, , , ), 
    Ψ_1n()=   ∑_i=1^nψ_1(_i, ),

where _i()=(_1i(_1i_1≥ 0), ⋯, _Ti(_Ti_T≥ 0)). For ease of expression, we denote ψ^(m)(, , , )=()ψ_0^(m)(, , , ), and ψ_0^(m)(_i, )=ψ_0^(m)(_i, , , ), where ψ_0^(m)(_i, ) is free of , and ψ^(m)(, , , ) satisfies ψ^(m)(, , , )=0 with true parameters _0, _0, and _0. We propose the integrated statistic with ω̃_ij defined in (<ref>), i.e.,

    T_n^(m) = 1/n(n-1)∑_i≠ jω̃_ijψ_0^(m)(_i, )ψ_0^(m)(_j, ).


Let ^t()=∂[_ti(_ti_1≥0)ψ_0^(m)(, _0)]/∂, 
()=(^1(), ⋯, ^T()),
and the kernel of a U-statistic under the null hypothesis be

    h^(m)(_i, _j)
    		=   ω̃_ijψ_0^(m)(_i, _0)ψ_0^(m)(_j, _0)- ψ_1(_i, _0)_j
       - _iψ_1(_j, _0) + ψ_1(_i, _0)ψ_1(_j, _0),

where

    _i
    	=∑_t=1^T∫__t∈Θ_θ_t
    	J^t()_ti(_ti_t≥0)ψ_0^(m)(_i, _0) w_t(_t)d_t

and

    =∑_t=1^T∫__t∈Θ_θ_t
    	J^t()^t()J w_t(_t)d_t.


We are now ready to derive the asymptotic distributions of T_n^(m) under the null and local alternative hypotheses.

	If Assumptions (A1)–(A5) hold, then under the null hypothesis, we have
	
    nT_n^(m) -μ_0 ζ,

	where μ_0=-2[ψ_1(_1, _0)_1]
	+[ψ_1(_1, _0)ψ_1(_1, _0)], ζ is a random variable of the form ζ=∑_j=1^∞λ_j(χ^2_1j-1), and χ^2_11, χ^2_12, ... are independent χ^2_1 variables.
	Here, {λ_j} are the eigenvalues of the kernel h^(m)(_1, _2) under f(, _0, , _0), i.e., they are the solutions of λ_jg_j(_2)=∫_0^∞h^(m)(_1, _2)g_j(_1)f(_1, _0, , _0)d_1 for nonzero g_j.



Proof. 
	We show this Theorem as the same as the proof of Theorem 1.
	Define notations as follows.
	
    =   (
    		[ ∫__1∈Θ_θ_1[_1i(_1i_1≥0)∂ψ_0^(m)(_i, _0)/∂]w(_1) d_1;                                                   ⋮; ∫__T∈Θ_θ_T[_Ti(_Ti_T≥0)∂ψ_0^(m)(_i, _0)/∂]w(_T) d_T ]),
    _i =   (
    		[ ∫__1∈Θ_θ_1_1i(_1i_1≥0)w(_1) d_1;                               ⋮; ∫__T∈Θ_θ_T_Ti(_Ti_T≥0)w(_T) d_T ]),
    _ij() =   ω̃_ijψ_0^(m)(_i, )ψ_0^(m)(_j, ).

	By Assumption (A1), we have
	
    [()ψ_0^(m)(, )]
    		=   [()ψ_0^(m)(, _0)]	- ()JΨ_1n(_0)+o_p(n^-1/2),

	and
	
    1/n∑_i≠ j_ij()
    		=   1/n∑_i≠ jω̃_ijψ_0^(m)(_i, )ψ_0^(m)(_j, )
    
    		=   1/n∑_i≠ j[_ij(_0) - 1/nΨ_1n(_0)_i- 1/n_jΨ_1n(_0) + 1/n^2Ψ_1n(_0)Ψ_1n(_0)]
       +1/n∑_i≠ j[_i	- 1/n JΨ_1n(_0)] o_p(n^-1/2)
       +1/n∑_i≠ j[_j-1/n JΨ_1n(_0)] o_p(n^-1/2)+o_p(n^-1).

	By notations, we have, for any i≠ j,
	
    [_i] =     [_ij(_0)] = 0.


	It can be shown by the central limit theorem that
	
    1/n∑_i≠ j[_i	- 1/n JΨ_1n(_0)]
    		=n-1/n∑_i=1^n[_i	- 1/n JΨ_1n(_0)]
    		=O_p(n^1/2).

	which implies that
	
    T_n =   1/n(n-1)∑_i≠ j_ij()
    
    		=   1/n(n-1)∑_i≠ j[_ij(_0) - 1/nΨ_1n(_0)_i- 1/n_jΨ_1n(_0) + 1/n^2Ψ_1n(_0)Ψ_1n(_0)]
       +o_p(n^-1)
    ≡   T_n1+T_n2+T_n3+o_p(n^-1),

	where
	
    T_n1 =   1/n(n-1)∑_i≠ j_ij(_0),
    
    		T_n2 =    -2/n^2∑_i=1^n Ψ_1n(_0) K_i,
    
    		T_n3 =   1/n^2Ψ_1n(_0) HΨ_1n(_0).

	As the same arguments as the proof of Theorem 2, we have
	
    T_n - μ_0/n = _n + R_n +o_p(n^-1),

	where
	
    _n = 1/n(n-1)∑_i≠ jh(_i,_j)

	with the kernel given in (22) in the main paper
	
    h^(m)(_i,_j)
    			=   ω̃_ijψ_0^(m)(_i, _0)ψ_0^(m)(_j, _0)- ψ_1(_i, _0)_j
       - _iψ_1(_j, _0) + ψ_1(_i, _0)ψ_1(_j, _0),

	and R_n=o_p(n^-1).

	According to Theorem in 5.5.2 of <cit.> by checking it conditions, we can show that
	
    n_n ζ,

	where ζ is a random variable of the form ζ=∑_j=1^∞λ_j(χ^2_1j-1), and χ^2_11,χ^2_12,... are independent χ^2_1 variables, that is, ζ has characteristic function
	
    E[e^itζ]=∏_j=1^∞(1-2itλ_j)^-1/2e^-itλ_j.



	If Assumptions (A1)–(A6) hold, then under the local alternative hypothesis that is =n^-1/2 with a deterministic vector ∈Θ_β, we have
	
    nT_n^(m)-μ_0 ζ,

	where μ_0 is defined in Theorem <ref>, ζ is a random variable of the form ζ=∑_j=1^∞λ_j(χ^2_1j(μ_aj)-1), and χ^2_11(μ_a1), χ^2_12(μ_a2), ⋯ are independent noncentral χ^2_1 variables.
	Here, {λ_j} are the eigenvalues of the kernel h^(m)(_1, _2) defined in (<ref>) under f(, _0, , _0), i.e., they are the solutions of λ_jg_j(_2)=∫_0^∞h^(m)(_1, _2)g_j(_1)f(_1, _0, , _0)d_1 for nonzero g_j, and each noncentrality parameter of χ^2_1j(μ_aj) is
	
    μ_aj = [ϕ_j(_0)∂log(f(_0, _0, , _0))/∂],    j=1, 2, ⋯,

	where {ϕ_j()} denote orthonormal eigenfunctions corresponding to the eigenvalues {λ_j}, and _0 is generated from the null distribution f(, _0, , _0).


The proof of this theorem can be completed along the same lines of Theorem 2 and Theorem <ref>.

Next, we demonstrate the finite-sample performance of the proposed WAST in multiple change-plane models. 
Consider the GLM in example 2 with the canonical parameter

    μ =  + _1(_1_1≥0)+_2(_2_2≥0),

where  is the same as that in Section 5 in the main paper, _1=_2 = τ_q, and , , and _1=_2 are generated as in Section 5 in the main paper. Let _-1,s be _s excluding the first component, and let _-1,s be _s excluding the first component, s=1,2. θ_11 and θ_21 are chosen as the negatives of the 0.3 and 0.6 percentiles of _-1,1_-1,1 and _-1,1_-1,1, respectively, which means that _1_1 divides the population into two groups with 0.7 and 0.3 observations, and _1_1 divides the population into two groups with 0.4 and 0.6 observations. When (r,p,q)=(2,2,3) or (r,p,q)=(6,6,3), the maximal value of the range κ_max is log(0.7), log(0.35) and log(0.7) for Gaussian, binomial and Poisson regression, respectively. While (r,p,q)=(2,2,11) or (r,p,q)=(6,6,11), the maximal value of the range κ_max is log(0.9), log(0.75), and log(0.9) for Gaussian, binomial, and Poisson regression, respectively. The other settings are the same as in Section 5 in the main paper.

Table <ref> lists the sizes of the proposed WAST with multiple change planes; as can be seen, the sizes are close to the nominal level of 0.05, as expected. Figure <ref> shows the power of the proposed WAST with multiple change planes; as can be seen, all the powers (i) increase as the sample size increases and (ii) increase rapidly as  departs from the null hypothesis.







 §.§ Change plane analysis for quantile, probit and semiparametric models

For fair comparison, in <ref>–<ref>, we consider the settings as similar to the existing literature <cit.>, which are different from the GLMs in Section 5 in the main paper.

We consider the probit regression model in the main paper

    f(_i, , , )=Φ(0.5+_1i+_i(_i≥0))^Y_iΦ(-0.5-_1i-_i(_i≥0))^1-Y_i.

We generate X_1i independently from binomial distribution b(1,0.5), _i from p-variate normal distribution with mean  and variance √(2)I, and Z_1i=1 and (Z_2i,⋯,Z_qi) from (q-1)-variate standard normal distribution.
We set =(1,⋯,1) under H_0, and (θ_2, ⋯,θ_q) = (1,2,⋯,2) under H_1, where θ_1 is chosen as the negative of the 0.65 percentile of Z_2θ_2+⋯+Z_qθ_q, which means that  divides the population into two groups with 0.35 and 0.65 observations, respectively.

Consider the quantile regression model in the main paper,

    Y_i = 0.5+_i+_i(τ)(_i(τ)≥0) + _i,

where _i is generated from standard normal distribution, _i and _i are generated as same as these in probit regression model. The error _i is generated from standard normal distribution.
The parameters  under H_0 and  under H_1 are same as these in probit model.


Consider the semiparametric model in the main paper,

    Y_i = γ(_i,_2) +_i A_i (_i≥0)+_i.

As the same settings provided by <cit.>, we consider two settings for the propensity score model π(_i,_1), denoted by P1 and P2:

	
  * π(_i,_1)=0.5;
	
  * π(_i,_1)=exp(0.5_2i+0.5_3i)/1+exp(0.5_2i+0.5_3i),

and two baseline mean functions for γ(_i,_2), denoted by B1 and B2:

	
  * γ(_i,_2) = 1+0.5_2i+_3i^2;
	
  * γ(_i,_2) = 1+sin(_2i+_3i).

We calculate the WAST and SST in the main paper by fitting a linear model for baseline function γ(_i,_2) and a logistic regression for  the propensity score model π(_i,_1). Therefore, two baseline models γ(_i,_2) are misspecified. We generate v_1i independently from binomial distribution b(1,0.5), v_2i independently from uniform distribution U(-1,1), and X_ji independently from uniform distribution U(-1,1) , where i=1,⋯,n and j=1,⋯,p. We set _i=(1, v_1i, v_2i) and _i=(1, v_1i, v_2i, X_1i,⋯,X_(p-3)i).
The parameters  under H_0 and  under H_1 are same as these in probit model.

For all models with change plane analysis, we evaluated the power under a sequence of alternative models indexed by κ, that is H_1^κ: ^κ=κ^* with κ = i/10 for semiparametric model and κ=i/20 for others, i=1,⋯,10, where ^*=(1,⋯,1). We set sample size n=(200, 400, 600), 1000 repetitions and 1000 bootstrap samples, and report in Figure <ref>-<ref> the performance for both the WAST and SST. We calculate SST over {^(k)=(θ^(k)_1,⋯,θ^(k)_q): k=1,⋯,K} with the number of threshold values K=2000. Let ^(k)_-1=^(k)_-1/^(k)_-1, where ^(k)_-1=(θ^(k)_2,⋯,θ^(k)_q) and ^(k)_-1=(θ̃^(k)_2,⋯,θ̃^(k)_q), and ^(k)_-1 is drawn independently form (r-1)-variate standard normal distribution.  For each θ_1^(k), k=1,⋯,K, we selected it by equal grid search in the range from the lower 10th percentile to upper 10th percentile of the data points of {θ^(k)_2Z_2i+⋯+θ^(k)_qZ_qi}_i=1^n, which is same as that in <cit.>. Here we consider four combinations of (p,q)=(1, 3), (5, 5), (10, 10), (50, 20).

Type 1 errors (κ=0) with sample size n=(200, 400, 600) are list in Table <ref>. We can see from Table <ref> that the size of the proposed WAST are close to the nominal significance level 0.05, but for most scenarios the size of the SST are much smaller than 0.05.
Figure <ref>-<ref> indicate that powers become greater as sample size n increases, which are verified the asymptotic theory. The proposed WAST has comparable power with the SST for the semiparametric model, but the size of the SST is much less than the nominal level 0.05 in Table <ref>. SST has wrong size, and is less powerful than the WAST for probit and quantile regression models in four combinations of (p,q). Since the size is much less than the nominal level 0.05 when dimension of  not smaller than 5 in Table <ref>, we only depict in Figure <ref> the power curves for the WAST. Figure <ref> shows that the power increases fast as the sample size becomes large as expected from the theory, and the power increases rapidly in all models and two combinations (p,q)=(10, 10) and (50, 20) as the alternative is farther away from the null.



















 §.§ Change plane analysis for quantile, probit and semiparametric models with  from 

We consider probit regression, quantile regression and semiparametric models with same settings as Section <ref> in the main paper but different   and error's distribution of quantile regression model. We generate Z_ji independently from normal distribution with mean zero and standard deviation 5, and set _i=(1,Z_1i,⋯,Z_(q-1)i), where j=1,⋯,q-1. For quantile regression, we generate error _i independently from t_3 distribution with degrees freedom 3.

Type 1 errors of the proposed test statistic are listed in Table <ref>, and the power curves are depicted in Figure <ref>-<ref>. From Table <ref>, we can see that Type 1 errors of the proposed test statistic are all close to the nominal significant level 0.05, while that of the SST are much smaller the 0.05 except for the semiparametric model with B2 plus P2.
We can find from Figure <ref>-<ref> that the power of the proposed test statistic increases much faster than that of SST.
















 §.§ Change plane analysis for quantile, probit and semiparametric models with  from  distribution

We consider probit regression, quantile regression and semiparametric models with same settings as Section <ref> in current Supplementary Material but different   and error's distribution of quantile regression model. We generate Z_ji independently from t_3 distribution with 3 degrees of freedom, and set _i=(1,Z_1i,⋯,Z_(q-1)i), where j=1,⋯,q-1. For quantile regression, we generate error _i independently from t_3 distribution with degrees freedom 3.

Type 1 errors of the proposed test statistic are listed in Table <ref>, and the power curves are depicted in Figure <ref>-<ref>. We have the similar conclusion from Table <ref> and Figure <ref>-<ref>.
















 §.§ Quantile regression with Cauchy distribution

In this Section, we consider the quantile regression with error from standard Cauchy distribution. Other settings are same as Section <ref> in current Supplementary Material.

Type 1 errors of the proposed test statistic are listed in Table <ref>, and the power curves are depicted in Figure <ref>-<ref>. We have the similar conclusion from Table <ref> and Figure <ref>-<ref>.















§ CASE STUDIES


D.figure



 §.§ A histogram plot

The histogram plot of the support incomes for the elderly in Shanxi Province of China is presented in this section. From this plot, we can observe that the support income has a heavy-tailed distribution.





 §.§ Boston housing dataset

We apply the proposed test to the Boston housing dataset from <cit.>, which includes 14 variables that were measured across 506 census tracts in the Boston area. 
The response variable is the logarithm of the median value (MV) of the houses in those census tracts of the Boston Standard Metropolitan Statistical Area. 
The meaning of the other 13 social-economic, structural, and regional variables is as follows:
(1) Crime rate by town (CRIM);
(2) Proportion of residential land with a floor area of over 25,000 square feet (ZN);
(3) Proportion of nonretail business in each town (INDUS);
(4) Charles River dummy variable coding 1 if tract bounds the river and 0 otherwise (CHAS);
(5) Nitrogen oxide concentration in parts per 10 million (NOX);
(6) Average number of rooms per dwelling (RM);
(7) Proportion of self-occupied units built before 1940 (AGE);
(8) Weighted distances to five employment centers in Boston (DIS);
(9) Index of accessibility to radial highways (RAD);
(10) Full-value property-tax rate per $10,000 (TAX);
(11) Pupil/teacher ratio by town (PTRATIO);
(12) B = 1000( Bk - 0.63)^2, where Bk is the proportion of black people by town (B); and
(13) Percentage of people with lower status (LSTAT);

As explored in <cit.>, the interest was to analyze the impact of air quality on the median housing costs. 
Thus, we take NOX as  for our analysis. Furthermore, based on the regression tree depicted in Figure 1 of <cit.>,  LSTAT was identified as the root node with two descendent nodes CRIM and RM. Therefore, we take CRIM, RM and LSTAT as the grouping variables . 
The baseline covariate  includes seven variables: NOX, ZN, CHAS, DIS, RAD, TAX, and B. 
We consider the linear change-plane regression model, i.e.,

    Y_i = _i + _i(_i≥0)+_i,    i=1,⋯,506,

with _i=(1,NOX_i,ZN_i,CHAS_i,DIS_i,RAD_i,TAX_i,B_i), _i=(1,CRIM_i, RM_i, LSTAT_i), and _i=NOX_i.

The p-value of the WAST is less than 0.0001, which is calculated based on 5000 bootstrap samples. 
Therefore, this gives a strong evidence for rejecting the null hypothesis, that is, there exists a subgroup in which the air quality has different effects on the median value of the houses.

We, then, estimate parameters (,,) using a similar way as in the previous subsection. 
The estimators are =(3.463,-0.836,0.001,0.157,-0.025,0.004,-0.001,0.001), 
 =1.002, and =(-0.986,-0.009,0.166,-0.012).
This gives one of subgroups, denoted by G^+={i:^T_i≥ 0}, 
    G^+={i: -0.009*CRIM+0.166*RM-0.012*LSTAT≥ 0.986}.
 
There are 132 census tracts in the subgroup G^+. From the estimation results, it is observed that the NOX variable has a positive influence on the MV of the houses in the subgroup G^+, while conversely, it exerts a negative impact on MV in another subgroup. 

To demonstrate the impact of NOX on MV in different groups, we plot the regression lines in <ref>. 
Based on <ref> (a), it is evident that in the subgroup G^+, the median value of houses increases as the air quality decreases, while as shown in <ref> (b), in another subgroup G^-={i:^T_i< 0}, the air quality has a positive impact on the median value of houses. This implies that in census tracts characterized by lower crime rates, higher room counts per dwelling, and smaller numbers of people with lower income, the median house value will still rise even in the presence of declining air quality. Thereby, combining <ref> (a) and (b), the existence of significant  heterogeneity further supports our testing findings that there are subgroups.






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