In many applications, researchers are interested in the direct and indirect causal effects of an intervention on an outcome of interest. Mediation analysis offers a rigorous framework for the identification and estimation of such causal quantities. In the case of binary treatment, efficient estimators for the direct and indirect effects are derived by <cit.>. These estimators are based on influence functions and possess desirable multiple robustness properties. However, they are not readily applicable when treatments are continuous, which is the case in several settings, such as drug dosage in medical applications. In this work, we extend the influence function-based estimator of <cit.> to deal with continuous treatments by utilizing a kernel smoothing approach. We first demonstrate that our proposed estimator preserves the multiple robustness property of the estimator in <cit.>. Then we show that under certain mild regularity conditions, our estimator is asymptotically normal. Our estimation scheme allows for high-dimensional nuisance parameters that can be estimated at slower rates than the target parameter. Additionally, we utilize cross-fitting, which allows for weaker smoothness requirements for the nuisance functions.

$*$ denotes equal contribution.

§ INTRODUCTION

Estimating the effect of a treatment, policy or intervention is of interest in various fields such as epidemiology, economics, medicine and sociology. A common estimand is the Average Causal Effect (ACE), which has been extensively studied in the literature <cit.>. However, in addition to estimating the treatment effect, one can also be interested in the pathways and mechanisms through which the treatment affects the outcome of interest. Causal mediation analysis offers a precise and rigorous mathematical framework to answer such questions. Causal mediation analysis has been explored in depth in the literature, see <cit.>.

Much of the literature on mediation analysis assumes binary treatments. However, interventions involving the dosage of a drug, the duration of an activity, or the frequency of activity are better described with continuous treatments. In such cases, mediation effects are naturally described by a multi-dimensional surface rather than a scalar parameter. This scenario is challenging because it involves learning a multi-dimensional surface without imposing a priori shape constraints. Additionally, in the presence of high dimensional nuisance parameters, the estimator may inherit the slow rates of our nuisance estimators, adversely affecting inference for the target parameter.

The challenges related to estimating ACE in the continuous treatment setting have been addressed by <cit.>. A common method for dealing with continuous treatments involves using Bayesian Additive Regression Trees (BART) as used by <cit.>. However, this requires correct specification of the relevant models, and inherits the rate of the outcome regression estimation. An alternative approach that leverages semiparametric theory involves specifying a parametric form for the dose response curve, or projecting the true curve onto a parametric model as presented by <cit.>. However, these methods may suffer from bias under misspecification of the dose response curve.

In contrast to approaches involving parametric assumptions on the dose response curve, flexible approaches to modeling the dose response curve have also been proposed. For example, <cit.> utilize a two-stage estimator that first constructs a doubly robust pseudo-outcome in the first stage and then regresses the pseudo-outcome on the treatment in the second-stage using non-parametric regression methods. <cit.> utilize double machine learning along with applying kernel  smoothing to the Augmented Inverse Propensity Weighted (AIPW) score <cit.>. This allows for slower estimation of nuisance parameters while still obtaining fast rates for the target parameter. However, dealing with continuous treatments in mediation analysis has not been studied to the same extent.

In this paper we propose a kernel smoothing approach combined with influence function based estimators <cit.> to deal with continuous treatment for causal mediation analysis. We propose an estimator that, under mild regularity conditions, is consistent, asymptotically linear and well as asymptotically normal. Our work aims to extend the results for continuous treatment ACE for the case of mediation analysis involving continuous treatments in the presence of high-dimensional covariates. <cit.> tackle this problem by weighting the observations by a generalized propensity score that is given as either the conditional density of treatment given (1) the covariates or (2) the covariates and the mediator. The authors estimate the generalized propensity score either parametrically or non-parametrically, and establish asymptotic normality. However, this method inherits the rate of estimation of the generalized propensity score, which can be slow. In contrast, we propose an approach motivated by influence functions and hence obtain many of the desirable properties of influence functions, namely allowing for slower estimation of nuisance parameters, as well as robustness properties. Our work draws heavily from the existing causal mediation literature discussing the identification and estimation of such effects <cit.>. Additionally, we utilize the double machine learning paradigm from <cit.>.

The rest of the paper is organized as follows. Section 2 introduces the formal mediation framework, describes the identifying assumptions, and discusses an influence function based estimator for binary treatments. Section 3 extends the influence function-based approach to continuous treatment settings, describing the sample-splitting procedure and smoothing procedure. In Section 4, we provide our main result, along with the requisite regularity conditions.

§ MEDIATION ANALYSIS

Let $A$ be the continuous treatment variable taking values in $\mathcal{A}$, $Y$ be the outcome variable with values in $\mathcal{Y}$, and $M$ be a mediator variable with values in $\mathcal{M}$, which relays parts of the causal effect of $A$ to $Y$. Also, let $X$ denote observed pre-treatment covariates in the system taking values in $\mathcal{X}$.
In order to desribe the causal effect of the treatment on the outcome, we use the potential outcome framework <cit.>. Let $Y^{(A=a)}$ be the potential outcome variable representing the outcome had (contrary to the fact) the treatment is set to value $a$.
Suppose we are interested in changing the value of the treatment from $a$ to $a'$. A popular way to measure the causal effect of this change of treatment is to use the average causal effect (ACE), which captures the difference in the expected value of the counterfactual outcome variables, that is

\[
\]

where $\mathbb{E}[\cdot]$ denotes the population-level expectation operator.

The total average causal effect of the treatment on the outcome $Y$ can be partitioned into the part that is mediated by the variable $M$, and the rest of the causal effect.
To formally define this partitioning, let $Y^{(a,m)}$ denote the potential outcome variable corresponding to the outcome had the treatment is set to value $a$ and the mediator is set to value $m$, and $M^{(a)}$ denote the mediator variable had the treatment is set to value $a$. <cit.> proposed the following partitioning of the average causal effect into the natural direct and indirect effects:

\begin{equation}
\label{eq:NDENIE}
\begin{aligned}
&=\overbrace{\mathbb{E}[Y^{(a)}-Y^{(a')}]}^\text{total effect}\\
&=\underbrace{\mathbb{E}[Y^{(a,M^{(a)})}-Y^{(a,M^{(a')})}]}_\text{natural indirect effect}
+\underbrace{\mathbb{E}[Y^{(a,M^{(a')})}-Y^{(a',M^{(a')})}]}_\text{natural direct effect}.
\end{aligned}
\end{equation}

In words, the quantities  natural direct effect (NDE) and  natural indirect effect (NIE) can be described as follows. NDE captures the change in the expectation of the outcome if the value of the treatment variable is switched between the two arms of the experiment, while the mediator behaves as if the treatment had not changed. NIE captures the change in the expectation of the outcome if the value of the treatment variable is fixed, while the mediator behaves as if the treatment had been switched between the two arms of the experiment.
In the following subsection, we discuss estimating NDE and NIE from observational data.

§.§ Estimating Natural Direct and Indirect Effects

In order to estimate the natural direct and indirect effects, from the partitioning in display (<ref>) it is clear that it suffices to focus on estimating quantities of the form

\[
\psi_0(a,a')=\mathbb{E}[Y^{(a,M^{(a')})}],
\]

for $a,a'\in\mathcal{A}$. Suppose i.i.d. data from distribution $P$ on variables $O=\{A,X,M,Y\}$ is given.
In general, the estimand $\psi_0$ is not identified from observational data and identification assumptions are needed to relate the distribution of the observational data to that of  counterfactual variables.
We required the following assumptions for the identification

[Identification Assumptions]

* Consistency. For all $a\in\mathcal{A}$ and $m\in\mathcal{M}$,

\begin{align*}
Y^{(a,m)}=Y&\quad\text{almost surely if }A=a\text{ and } M=m,\\
M^{(a)}=M&\quad\text{almost surely if }A=a.
\end{align*}

* Sequential Exchangeability. For all $a,a'\in\mathcal{A}$,

\begin{align*}
&(Y^{(a,m)},M^{(a')})\perp A~|~X,\\
&Y^{(a,m)}\perp M~|~A=a',X.
\end{align*}

* Positivity. For all $a\in\mathcal{A}$ and $m\in\mathcal{M}$,

\begin{align*}
&f_{M|A,X}(m|A,X)>0\quad\text{almost surely},\\
&f_{A|X}(a|X)>0\quad\text{almost surely},
\end{align*}

where $f_{M|A,X}$ and $f_{A|X}$ are the conditional density of $M$ given $A$ and $X$, and the conditional density of $A$ given $X$, respectively.

Under Assumption <ref>, the estimand $\psi_0(a,a')$ can be identified using the following expression called the mediation functional, which was originally proposed in <cit.>.

\begin{equation}
\label{eq:MCF}
\psi_0(a,a')=\int_\mathcal{M}\int_\mathcal{X}\mathbb{E}[Y|A=a,M=m,X=x]f_{M|A,X}(m|A=a',X=x)f_X(x)d\mu(m,x),
\end{equation}

where, $f_X$ is the marginal distribution of $X$, and $\mu$ is a dominating measure for the distribution of $(M,X)$.

Using the expression (<ref>), one can estimate the parameter of interest $\psi_0(a,a')$ by first estimating the nuisance functions $\mathbb{E}[Y|A,M,X]$ and $f_{M|A,X}$, and then using a plug-in estimator to estimate $\psi_0$ as follows

\[
\hat{\psi}^{MF}_0(a,a')=\frac{1}{n}\sum_{i=1}^n
\int_\mathcal{M}\hat{\mathbb{E}}[Y_i|A=a,M=m,X_i]\hat{f}_{M|A,X}(m|A=a',X_i)d\mu(m).
\]

Unfortunately, this estimator is sensitive to bias in the estimation of the nuisance functions, that is, misspecification of either of the nuisance functions induces bias in the estimation of the parameter of interest.

As an alternative approach, for the case of binary treatments, i.e., $\mathcal{A}=\{0,1\}$, <cit.> developed a general semiparametric framework for obtaining inferences about the parameter $\psi$ in the nonparametric model with unrestricted observed data model.
The authors derived the efficient influence function for $\psi_0(a,a')$

\begin{equation}
\label{eq:IFb}
\begin{aligned}
&= \frac{I(A = a)f_{M|A,X}(M | A = a', X)}{f_{A|X}(a | X)f_{M|A,X}(M | A = a, X)}\{Y - \mathbb{E}[Y | A = a,M,X]\}\\
&\quad+ \frac{I(A = a')}{f_{A|X}(a' | X)}\{\mathbb{E}[Y | A = a,M,X] - \eta(a, a', X)\} + \eta(a, a', X) - \psi_0(a,a'),
\end{aligned}
\end{equation}

where $a,a'\in\{0,1\}$, $I(\cdot)$ denotes the indicator function, and

\[
\eta(a, a', X) = \int_\mathcal{M} \mathbb{E}[Y | A = a, M = m,X]f_{M|A,X}(m | A = a', X) d\mu(m).
\]

Note that $IF_{\psi_0}$ is a function of three nuisance functions $\mathbb{E}[Y|A,M,X]$, $f_{M|A,X}$, and $f_{A|X}$.
<cit.> showed that the estimator based on this influence function has the triple robustness property. That is, it will be consistent even if the model for one (but not more than one) nuisance function is misspecified. Formally, let

* $\mathfrak{M}_{ym}$ be the sub-model in which the model for $\mathbb{E}[Y|A,M,X]$ and $f_{M|A,X}$ are correctly specified.

* $\mathfrak{M}_{ya}$ be the sub-model in which the model for $\mathbb{E}[Y|A,M,X]$ and $f_{A|X}$ are correctly specified.

* $\mathfrak{M}_{ma}$ be the sub-model in which the model for $f_{M|A,X}$ and $f_{M|A,X}$ are correctly specified.

Then the estimator for $\psi_0$ based on the influence function $IF_{\psi_0}$ is consistent in the submodel $\mathfrak{M}_{ym}\cup\mathfrak{M}_{ya}\cup\mathfrak{M}_{ma}$.

Our goal is to extend the theory of <cit.> to the case of continuous treatment variables. This is the objective in the following section.

§ CONTINUOUS MEDIATION ANALYSIS

In this section we show that identification results closely related to the results of <cit.> can be established for the case of continuous treatments.
The weights in the moment function (<ref>) contain indicator functions which is a problematic feature when dealing with continuous treatments. We modify the weights by utilizing kernel smoothing technique, in which the data with treatment value in a neighborhood defined by bandwidth parameter $h$ is used for weighting each point.

Let $d_A$ denote the dimension of the treatment variable, and let

\[
\]

where $k(\cdot)$ is a kernel function, and $h$ denotes the bandwidth parameter.
We propose the following moment function for estimating $\psi_0(a,a')$.

\begin{equation}
\label{eq:IF}
\begin{aligned}
&= K_h(A - a)\lambda(a,X)\frac{\alpha(a',M,X)}{\alpha(a,M,X)}\{Y - \gamma(a,M,X)\}\\
&\quad+ K_h(A - a')\lambda(a',X)\{\gamma(a,M,X) - \eta(a, a', X)\} + \eta(a, a', X) - \psi(a,a'),
\end{aligned}
\end{equation}

where $\lambda(a,X) := 1/{f_{A|X}(a|X)}$, $\alpha(a,M,X) := f_{M|A,X}(M| a, X)$, and $\gamma(a,M,X) := \mathbb{E}[Y|A=a,M,X]$ are the nuisance functions.

We require the kernel function to satisfy the following conditions.
[Kernel & Bandwidth Assumptions]

the kernel function $k(\cdot)$ satisfies

* $\int k(u) du = 1$

* $\int uk(u) du = 0$

* $0 < \int u^2 k(u) du < \infty$

* Bounded Differntiable: $\left|\frac{dk(u)}{du} \right| < C |u|^{-v}$

* $\int K_h^2(u) du < \infty$

Additionally, the kernel bandwidth is assumed to be a function of the sample size $n$ which satisfies $h \rightarrow 0$, $nh^{d_A} \rightarrow \infty$ and $nh^{d_A + 4} \rightarrow C$, for a constant $C$, as $n$ goes to infinity.

Note that in the moment function (<ref>), the nuisance parameters are not functions of the parameter of interest $\psi$. Therefore, it gives us the important property that we do not need to estimate the entire law and having estimators for nuisance functions suffices for obtaining an estimator for the parameter of interest. That is, given estimations of nuisance functions $\hat{\alpha},\hat{\lambda},\hat{\gamma}$, the parameter of interest can be estimated as $\hat{\psi}$ by solving the following estimating equation

\[
\mathbb{E}[m(O;\hat{\alpha},\hat{\lambda},\hat{\gamma},\hat{\psi}(a,a'))]=0.
\]

§.§ Estimation Procedure

We use cross-fitting estimation approach of <cit.> for separating the estimation of the nuisance functions from the parameter of interest.
This approaches provides us with the benefit that weaker smoothness requirements are needed for the nuisance functions.
In the cross-fitting approach, we partition the samples into $L$ equal size  parts $\{I_1,...,I_L\}$.
For $\ell\in\{1,...,L\}$, we estimate the nuisance functions $\hat{\alpha}_\ell,\hat{\lambda}_\ell,\hat{\gamma}_\ell$ on data from all parts but $I_\ell$.
For all $\ell$, let $\hat{\psi}_\ell$ be the estimation of $\psi_0$ obtained by solving

\[
\frac{1}{|I_\ell|}\sum_{i\in I_\ell}m(O_i;\hat{\alpha}_\ell,\hat{\lambda}_\ell,\hat{\gamma}_\ell,\hat{\psi}_\ell(a,a'))=0.
\]

Our final estimator of $\psi_0$ is obtained by

\begin{equation}
\label{eq:cf-estimator}
\hat{\psi}^{TR}(a,a')=\frac{1}{L}\sum_{\ell=1}^L \hat{\psi}_\ell(a,a').
\end{equation}

We next show that the proposed estimator preserves the triple robustness property of the approach of <cit.> when the treatments are continuous.

The estimator $\hat{\psi}^{TR}(a,a')$ is consistent in the submodel $\mathfrak{M}_{ym}\cup\mathfrak{M}_{ya}\cup\mathfrak{M}_{ma}$.

§ ASYMPTOTIC ANALYSIS

In this section, we study the asymptotic properties of the cross-fitting estimator in display (<ref>), and provide pointwise results.

We require the following regularity conditions for the results.

[Regularity Conditions]

* The conditional density functions $f_{A|X}$ and $f_{M|A,X}$ are finite and bounded away from zero.

* For all $M$ and $X$, the functions $\mathbb{E}[Y | a, M, X]f_{M|A,X}(M | a, X) f_{A|X}(a | X)$ and $f_{M|A,X}(M | a, X) f_{A|X}(a | X)$ as functions of $a$ are twice differentiable, with continuous second derivative, and the functions, and their first and second derivatives are bounded.

* The ground truth nuisance functions $\alpha,\lambda,\gamma$ and their estimations $\hat{\alpha},\hat{\lambda},\hat{\gamma}$ are bounded. Also, the estimation $\hat{\alpha}$ is bounded away from zero.

* $\|( [\frac{\partial^2}{\partial a^2} f_{M|A,X}(M | a, X) f_{A|X}(a | X)] )\gamma(a,M,X)\|_{\infty} < \infty$

* $\|\lambda(a)\left( \frac{\partial^2}{\partial a^2}[f_{M \mid A, X}(M \mid a, X) f_{A \mid X}(a \mid X)] \right) \|_{\infty} < \infty$

* $\|\frac{\partial^2}{\partial a^2} \left( f_{M \mid A, X}(M \mid a, X) f_{A \mid X}(a \mid X) \right)\|_{\infty} < \infty$

* $\|\frac{\partial^2}{\partial a^2} \left(\mathbb{E}[Y \mid M, a, X]f_{M \mid A, X}(M \mid a, X) f_{A \mid X}(a \mid X) \right) \|_{\infty} < \infty$

In addition to the regularity conditions, we require the following conditions regarding the convergence of the estimators of the nuisance functions.

[Convergence Conditions]

For any values $a$ and $a^\prime$, the estimators $\frac{\hat{\alpha}(a^\prime,\cdot,\cdot)}{\hat{\alpha}(a,\cdot,\cdot)},\hat{\lambda}(a,\cdot),\hat{\gamma}(a,\cdot,\cdot)$, and $\hat{\eta}(a,a^\prime,\cdot)$ are consistent, that is,

* $\int \left(\hat{\lambda}(a, \cdot) - \lambda(a, \cdot) \right)^2 f_{X}(X) dx \xrightarrow[]{P} 0$

* $\int_{\mathcal{X}} \int_{\mathcal{M}} \left(\frac{\hat{\alpha}(a^\prime, \cdot, \cdot)}{\hat{\alpha(a, \cdot, \cdot)}} - \frac{\alpha(a^\prime, \cdot, \cdot)}{\alpha(a, \cdot. \cdot)} \right)^2 f_{X}(X) dx dm \xrightarrow[]{P} 0$

* $\int_{\mathcal{X}} \int_{\mathcal{M}} \left(\hat{\gamma}(a, \cdot, \cdot) - \gamma(a, \cdot, \cdot) \right)^2 f_{X}(X) dx dm \xrightarrow[]{P} 0$

* $\int_{\mathcal{X}} \left(\hat{\eta}(a, a, \cdot) - \eta(a, a^\prime, \cdot)^2 \right) f_{X}(X) dx \xrightarrow[]{P} 0$

For any values $a$ and $a^\prime$, the estimators $\frac{\hat{\alpha}(a^\prime,\cdot,\cdot)}{\hat{\alpha}(a,\cdot,\cdot)},\hat{\lambda}(a,\cdot),\hat{\gamma}(a,\cdot,\cdot)$, and $\hat{\eta}(a,a^\prime,\cdot)$ are rate doubly robust, that is,

* $\sqrt{nh^{d_A}}\left(\int_{\mathcal{X}}\int_{\mathcal{M}}\left(\left[\frac{\hat{\alpha}(a^\prime, \cdot, \cdot)}{\hat{\alpha}(a, \cdot, \cdot)} - \frac{\alpha(a^\prime, \cdot, \cdot)}{\alpha(a, \cdot, \cdot)} \right]\right)^2f_{M, A, X}(M, a, X)dm dx\right)^{\frac{1}{2}}
\left(\int_{\mathcal{X}}\int_{\mathcal{M}} \left(\hat{\gamma}(a, \cdot, \cdot) - \gamma(a, \cdot, \cdot) \right)^2f_{M, A, X}(M, a, X) dm dx\right)^{\frac{1}{2}} \xrightarrow[]{P}0$

* $\sqrt{nh^{d_A}}\left(\int_{\mathcal{X}}\int_{\mathcal{M}}\left(\left[\hat{\lambda}(a, \cdot, \cdot) - \lambda(a, \cdot, \cdot) \right]\right)^2f_{A, X}(a, X)dm dx\right)^{\frac{1}{2}}
\left(\int_{\mathcal{X}}\int_{\mathcal{M}} \left(\hat{\gamma}(a, \cdot, \cdot) - \gamma(a, \cdot, \cdot) \right)^2f_{M, A, X}(M, a_, X) dm dx\right)^{\frac{1}{2}} \xrightarrow[]{P} 0$

* $\sqrt{nh^{d_A}}\left(\int_{\mathcal{X}}\int_{\mathcal{M}}\left(\left[\frac{\hat{\alpha}(a^\prime, \cdot, \cdot)}{\hat{\alpha}(a, \cdot, \cdot)} - \frac{\alpha(a^\prime, \cdot, \cdot)}{\alpha(a, \cdot, \cdot)} \right]\right)^2f_{M, A, X}(M, a, X)dm dx\right)^{\frac{1}{2}}
\left(\int_{\mathcal{X}}\int_{\mathcal{M}}\left(\left[\hat{\lambda}(a, \cdot) - \lambda(a, \cdot) \right]\right)^2f_{A, X}(a, X)dm dx\right)^{\frac{1}{2}}
\xrightarrow[]{P} 0$

* $\sqrt{nh^{d_A}}\left(\int_{\mathcal{X}}\int_{\mathcal{M}}\left(\left[\hat{\lambda}(a, \cdot) - \lambda(a, \cdot) \right]\right)^2f_{A, X}(a, X)dm dx\right)^{\frac{1}{2}}\left(\int_{\mathcal{X}}\int_{\mathcal{M}}\left(\left[\hat{\eta}(a, a^\prime, \cdot) - \eta(a, a^\prime, \cdot) \right]\right)^2f_{A, X}(a, X)dm dx\right)^{\frac{1}{2}}
\xrightarrow[]{P} 0$

Note that we do not require any convergence rates on individual nuisance functions.
The rate double-robustness condition implies that our requirements on the convergence rate is on the product of the functions. Therefore, if one of the functions converges at a slow rate, the other functions can compensate for that. This is an  desired property when working with non-parametric estimators that usually have slow convergence rates.

We have the following result regarding the convergence of the cross-fitting estimator.

Suppose Assumptions <ref>-<ref> hold. Then for any value pair $a,a'\in\mathcal{A}$, if $var()$ is bounded, then

\[
\sqrt{nh^{d_A}}(\hat{\psi}^{TR}(a,a')-\psi_0(a,a'))=
\sqrt{\frac{h^{d_A}}{n}}
\sum_{i=1}^nm(O_i;\alpha,\lambda,\gamma,\psi_0(a,a'))+o_p(1),
\]

and $\sqrt{nh^{d_A}}(\hat{\psi}^{TR}(a,a')-\psi_0(a,a')-B(a,a'))$ converges to the Gaussian distribution $\mathcal{N}(0,V(a,a'))$, with

\begin{align*}
B(a,a') = & h^2 \int u^2 k(u)du \\
&\times \mathbb{E}\Bigg\{ \frac{f(M \mid A = a', X)}{f(M \mid A = a, X)}\bigg(
\partial_a \gamma(X,M,a) \frac{\partial_a f(a|X,M)}{f(a \mid X)} + \frac{1}{2} \frac{f(a\mid X,M)}{f(a\mid X)}  \partial^2_a \gamma(X,M,a)
\bigg) \\
&+ \{\mathbb{E}[Y \mid X, M, A = a] - \eta(a, a', X)\}  \frac{1}{2} \frac{\partial^2_a f(a'|X,M)}{f(a'|X)} \Bigg\}+O(h^3),
\end{align*}

\begin{align*}
V(a,a')= & \int k(u)^2 du \times E_1 + O(h)
\end{align*}

where $E_1 =  \mathbb{E}\Bigg\{\frac{f(M \mid a', X)^2 f(a|X,M)}{f(M \mid a, X)^2f(a \mid X)^2} var(Y|X,M,a)+ \mathbb{E}\bigg\{\frac{1}{f(a'|X)}var[E(Y|X,M,a)|X,a']\Bigg\}$.

We next extend the result in Theorem <ref> to uniformity over a compact interior of the support of the treatment variable.
We require the following stronger version of Assumption <ref> for our result.

Let $\mathcal{A}_0$ be a compact interior of the support of the treatment variable. We assume that for the supremum of the pair $(a,a')\in\mathcal{A}_0$,
the consistency and rate double robustness conditions in Assumption <ref> hold. Moreover, the estimators $\frac{\hat{\alpha}(a',\cdot,\cdot)}{\hat{\alpha}(a,\cdot,\cdot)},\hat{\lambda}(a,\cdot),\hat{\gamma}(a,\cdot,\cdot)$, and $\hat{\eta}(a,a',\cdot)$ are Lipschitz continuous in $\mathcal{A}_0$.

Let the conditions in Theorem <ref> and Assumption <ref> hold. Then the asymptotically linear representation of in Theorem <ref> holds uniformly over $\mathcal{A}_0$.

§ CONCLUSION

In this paper we put forward a flexible method to estimate mediation effects in the presence of continuous treatments and high dimensional covariates. We provide results on rate as well as asymptotic normality to allow for the construction of confidence intervals and hypothesis tests. Extending the point-wise results to uniform results and providing optimal rates is left for future work.

§.§ Acknowledgment

We are grateful to Prof. Eric Tchetgen Tchetgen for helpful discussions and comments.

§ APPENDIX

§.§ Proof for Theorem 1

\begin{align*}
&\sqrt{nh^{d_A}} \frac{1}{n}\sum^L_{l=1}\sum_{i\in I_\ell} \Big\{m(O_i; \hat{\alpha}_\ell,\hat{\lambda}_\ell, \hat{\gamma}_\ell, \hat{\psi}_\ell(a,a'))- m(O_i;\alpha,\lambda,\gamma,\psi_0(a,a'))\Big\}\\
=&  \sqrt{\frac{h^{d_A}}{n}}\sum^L_{l=1}\sum_{i\in I_\ell} \Big\{m(O_i; \hat{\alpha}_\ell,\hat{\lambda}_\ell, \hat{\gamma}_\ell, \hat{\psi}_\ell(a,a'))-m(O_i;\hat{\alpha}_\ell,\hat{\lambda}_\ell,\hat{\gamma}_\ell,\psi_0(a,a'))\\
&+m(O_i;\hat{\alpha}_\ell,\hat{\lambda}_\ell,\hat{\gamma}_\ell,\psi_0(a,a'))- m(O_i;\alpha,\lambda,\gamma,\psi_0(a,a'))\Big\}\\
=& -\sqrt{nh^{d_A}} (\hat{\psi}^{TR}(a,a')- \psi_0(a,a') )\\ &+\sqrt{\frac{h^{d_A}}{n}}\sum^L_{l=1}\sum_{i\in I_\ell}\Big\{m(O_i;\hat{\alpha}_\ell,\hat{\lambda}_\ell,\hat{\gamma}_\ell,\psi_0(a,a'))- m(O_i;\alpha,\lambda,\gamma,\psi_0(a,a'))\Big\}\\
\end{align*}

from $\frac{1}{n}\sum^L_{l=1}\sum_{i\in I_\ell} m(O_i; \hat{\alpha}_\ell,\hat{\lambda}_\ell, \hat{\gamma}_\ell, \hat{\psi}_\ell(a,a'))=0$, there is

\begin{align*}
&\sqrt{nh^{d_A}} (\hat{\psi}^{TR}(a,a')- \psi_0(a,a') )\\
=& \sqrt{\frac{h^{d_A}}{n}}\sum^L_{l=1}\sum_{i\in I_\ell} \Big\{ m(O_i;\alpha,\lambda,\gamma,\psi_0(a,a'))\Big\}\\
&+\sqrt{\frac{h^{d_A}}{n}}\sum^L_{l=1}\sum_{i\in I_\ell}\Big\{m(O_i;\hat{\alpha}_\ell,\hat{\lambda}_\ell,\hat{\gamma}_\ell,\psi_0(a,a'))- m(O_i;\alpha,\lambda,\gamma,\psi_0(a,a'))\Big\}
\end{align*}

First, add and subtract $\frac{1}{|I_\ell|}\sum_{i\in I_\ell}m(O_i;\hat{\alpha}_\ell,\hat{\lambda}_\ell,\hat{\gamma}_\ell,\psi_0(a,a'))$ and using the definition of m-estimators, we need to demonstrate the following bound

\begin{align*}
\sqrt{\frac{h^{d_A}}{n}} \sum_{i \epsilon I_\ell} \left\{m(O_i;\hat{\alpha}_\ell,\hat{\lambda}_\ell,\hat{\gamma}_\ell,\psi_0(a,a')) - m(O_i;\alpha,\lambda,\gamma,\psi_0(a,a')) \right\} = o_p(1)
\end{align*}

The above can be broken down into the sum of the following terms and we show each is $o_p(1)$

\begin{align*}\tag{1}
\sqrt{\frac{h^{d_A}}{n}}\sum_{i \in \ell}\left\{ K_h(A_i - a)
\left[ \hat{\lambda}_i(a, X_i) - \lambda(a, X_i)\right]
\left[ \frac{\hat{\alpha}_i(a^\prime, M_i, X_i)}{\hat{\alpha}_i(a, M_i, X_i)} - \frac{ \alpha_i(a^\prime, M_i, X_i)}{\alpha_i(a, M_i, X_i)}\right]
\left[ Y_i - \gamma_i(a, M_i, X_i) \right]
\right\}
\end{align*}

\begin{align*}\tag{2}
\sqrt{\frac{h^{d_A}}{n}}\sum_{i \in \ell}\left\{K_h(A_i - a)
\left[\hat{\lambda}_i(a, X_i) - \lambda_i(a, X_i) \right]
\frac{\alpha(a^\prime, M_i, X_i)}{\alpha(a, M_i, X_i)}
\left[Y_i - \gamma_i(a, M_i, X_i)\right]
\right\}
\end{align*}

\begin{align*}\tag{3}
\sqrt{\frac{h^{d_A}}{n}}\sum_{i \in \ell} \left\{K_h(A_i - a)\lambda_i(a,X_i)
\left[\frac{\hat{\alpha}_i(a^\prime, M_i, X_i)}{\hat{\alpha}_i(a, M_i, X_i)} - \frac{\alpha_i(a^\prime, M_i, X_i)}{\alpha_i(a, M_i, X_i)}\right]
\left[Y_i  -  \gamma_i(a, M_i, X_i) \right]
\right\}
\end{align*}

\begin{align*}\tag{4}
\sqrt{\frac{h^{d_A}}{n}}\sum_{i \in \ell}\left\{-K_h(A_i - a)
\left[\hat{\lambda}_i(a, X_i) - \lambda_i(a, X_i) \right]
\left[\frac{\hat{\alpha}_i(a^\prime, M_i, X_i)}{\hat{\alpha}_i(a, M_i, X_i)} - \frac{\alpha_i(a^\prime, M_i, X_i)}{\alpha_i(a, M_i, X_i)} \right]
\left[\hat{\gamma}_i(a, M_i, X_i) - \gamma_i(a, M_i, X_i) \right]
\right\}
\end{align*}

\begin{align*}\tag{5}
\sqrt{\frac{h^{d_A}}{n}}\sum_{i \in \ell}\left\{-K_h(A_i - a)
\lambda_i(a, X_i)
\left[\frac{\hat{\alpha}_i(a^\prime, M_i, X_i)}{\hat{\alpha}_i(a, M_i, X_i)} - \frac{\alpha_i(a^\prime, M_i, X_i)}{\alpha_i(a, M_i, X_i)} \right]
\left[\hat{\gamma}_i(a, M_i, X_i) - \gamma_i(a, M_i, X_i) \right]
\right\}
\end{align*}

\begin{align*}\tag{6}
\sqrt{\frac{h^{d_A}}{n}}\sum_{i \in \ell}\left\{-K_h(A_i - a)
\left[\hat{\lambda}_i(a, X_i) - \lambda_i(a, X_i) \right]
\frac{\alpha_i(a^\prime, M_i, X_i)}{\alpha_i(a, M_i, X_i)}
\left[\hat{\gamma}_i(a, M_i, X_i) - \gamma_i(a, M_i, X_i) \right]
\right\}
\end{align*}

\begin{align*}\tag{7}
\sqrt{\frac{h^{d_A}}{n}}\sum_{i \in \ell}\left\{-K_h(A_i - a)
\lambda_i(a, X_i)
\frac{\alpha_i(a^\prime, M_i, X_i)}{\alpha_i(a, M_i, X_i)}
\left[\hat{\gamma}_i(a, M_i, X_i) - \gamma_i(a, M_i, X_i) \right]
\right\}
\end{align*}

\begin{align*}\tag{8}
\sqrt{\frac{h^{d_A}}{n}}\sum_{i \in \ell}\left\{K_h(A_i - a^\prime) \left[\hat{\lambda}_i(a^\prime, M_i, X_i) - \lambda(a^\prime, M_i, X_i) \right]
\left[\hat{\gamma}_i(a, M_i, X_i) - \gamma_i(a, M_i, X_i) \right] \right \}
\end{align*}

\begin{align*}\tag{9}
\sqrt{\frac{h^{d_A}}{n}}\sum_{i \in \ell}\left\{K_h(A_i - a^\prime) \left[\hat{\lambda}_i(a^\prime, X_i) - \lambda(a^\prime, X_i) \right]
\left[\gamma_i(a, M_i, X_i) \right] \right\}
\end{align*}

\begin{align*}\tag{10}
\sqrt{\frac{h^{d_A}}{n}}\sum_{i \in \ell}\left\{K_h(A_i - a^\prime) \lambda_i(a^\prime, X_i)
\left[\hat{\gamma}_i(a, M_i, X_i) - \gamma_i(a, M_i, X_i) \right]\right\}
\end{align*}

\begin{align*}\tag{11}
-\sqrt{\frac{h^{d_A}}{n}}\sum_{i \in \ell}\left\{K_h(A_i - a^\prime)\left[\hat{\lambda}_i(a^\prime, X_i) - \lambda(a^\prime, X_i ) \right]
\left[\hat{\eta}_i(a, a^\prime, X_i) - \eta(a, a^\prime, X_i) \right]\right\}
\end{align*}

\begin{align*}\tag{12}
-\sqrt{\frac{h^{d_A}}{n}}\sum_{i \in \ell}\left\{ K_h(A_i - a^\prime) \left[\hat{\lambda}_i(a^\prime, X_i) - \lambda(a^\prime, X_i) \right]\right\}
\eta(a, a^\prime, X_i)
\end{align*}

\begin{align*}\tag{13}
-\sqrt{\frac{h^{d_A}}{n}}\sum_{i \in \ell}\left\{K_h(A_i - a^\prime) \lambda(a^\prime, X_i)
\left[\hat{\eta}_i(a, a^\prime, X_i) - \eta(a, a^\prime, X_i) \right] \right\}
\end{align*}

\begin{align*}\tag{14}
\sqrt{\frac{h^{d_A}}{n}}\sum_{i \in \ell}\left\{\hat{\eta}_i(a, a^\prime, X_i) - \eta_i(a, a^\prime, X_i) \right\}
\end{align*}

$R1$, $R2$, $R3$ are $o_p(1)$

We prove the bound for $R1$, and $R2$ and $R3$ are bounded using a similar argument. To bound $R1$ we utilize the Conditional Markov Inequality and Lemma 6.1 from <cit.>. To bound R1, it suffices to show

\begin{align*}
\mathbb{E}\left[\sqrt{\frac{h^{d_A}}{n}}\sum_{i \in \ell}^n\left\{ K_h(A_i - a)
\left[ \hat{\lambda_i(a)} - \lambda(a)\right]
\left[ \frac{\hat{\alpha_i(a')}}{\hat{\alpha_i(a)}} - \frac{ \alpha_i(a')}{\alpha_i(a)}\right]
\left[ Y_i - \gamma_i(a) \right]
\right\}
\mid O_l^c \right] \xrightarrow[]{P} 0
\end{align*}

Since the data is i.i.d, we can instead bound

\begin{align*}
\sqrt{nh^{d_A}} \mathbb{E}\left[K_h(A_i - a)
\left[ \hat{\lambda_i(a, X_i)} - \lambda(a, X_i)\right]
\left[ \frac{\hat{\alpha_i(a^\prime, M_i, X_i)}}{\hat{\alpha_i(a, M_i, X_i)}} - \frac{ \alpha_i(a^\prime, M_i, X_i)}{\alpha_i(a, M_i, X_i)}\right]
\left[ Y_i - \gamma_i(a, M_i, X_i) \right]
\right]
\end{align*}

Split into two terms

\begin{align*}\tag{T1}
\sqrt{nh^{d_A}} \mathbb{E}\left[K_h(A_i - a)
\left[ \hat{\lambda_i(a, X_i)} - \lambda(a, X_i)\right]
\left[ \frac{\hat{\alpha_i(a^\prime, M_i, X_i)}}{\hat{\alpha_i(a, M_i, X_i)}} - \frac{ \alpha_i(a^\prime, M_i, X_i)}{\alpha_i(a, M_i, X_i)}\right] Y_i
\right]
\end{align*}

\begin{align*}\tag{T2}
- \sqrt{nh^{d_A}} \mathbb{E}\left[K_h(A_i - a)
\left[ \hat{\lambda_i(a, X_i)} - \lambda(a, X_i)\right]
\left[ \frac{\hat{\alpha_i(a^\prime, M_i, X_i)}}{\hat{\alpha_i(a, M_i, X_i)}} - \frac{ \alpha_i(a^\prime, M_i, X_i)}{\alpha_i(a, M_i, X_i)}\right] \gamma_i(a, M_i, X_i)
\right]
\end{align*}

Simplifying T2 T2 can be rewritten as

\begin{align*}
\sqrt{nh^{d_A}}\int_\mathcal{X}  \int_\mathcal{M} \int_\mathcal{A}
K_h(A_i - a)
\left[ \hat{\lambda}_i(a, X_i) - \lambda(a, X_i)\right]
\left[ \frac{\hat{\alpha}_i(a^\prime, M_i, X_i)}{\hat{\alpha}_i(a, M_i, X_i)} - \frac{ \alpha_i(a^\prime, M_i, X_i)}{\alpha_i(a, M_i, X_i)}\right] \gamma_i(a, M_i, X_i) f(M, A, X) da dm dx
\end{align*}

Applying integration by substitution, where $\frac{A_i - a}{h} = u$, the innermost integral can be rewritten as

\begin{align*}
\int_\mathcal{A} K_h(A_i - a)f(M \mid A, X) f(A \mid X) da
= \int_\mathcal{U} k(u)f(M \mid uh + a, X) f(uh + a \mid X) du
\end{align*}

A second-order Taylor expansion applied to $f(M \mid uh + a, X) f(uh + a \mid X)$ at $a$ gives

\begin{align*}
\int_\mathcal{U} k(u)\left( f(M \mid a, X) f(a \mid X) + uh \frac{\partial}{\partial a} f(M \mid a, X) f(a \mid X) + \frac{u^2h^2}{2}\frac{\partial^2}{\partial a^2} f(M \mid a, X) f(a \mid X) + o_p(h^2) \right)du\\
\end{align*}

A standard application of kernel algebra simplifies the above to

\begin{align*}
= f(M \mid a, X) f(a \mid X) + \frac{h^2}{2}\frac{\partial^2}{\partial a^2} f(M \mid a, X) f(a \mid X) \int u^2 k(u) du + o_p(h^2)
\end{align*}

Substituting this back into T2 gives

\begin{align*}
&\sqrt{nh^{d_A}}\int_\mathcal{X}  \int_\mathcal{M}
\left[ \hat{\lambda}_i(a, X_i) - \lambda(a, X_i)\right]
\left[ \frac{\hat{\alpha}_i(a^\prime, M_i, X_i)}{\hat{\alpha}(a, M_i, X_i)} - \frac{ \alpha_i(a^\prime, M_i, X_i)}{\alpha_i(a, M_i, X_i)}\right] \gamma_i(a, M_i, X_i)
f(M, a, X) dm dx\\
+ &\frac{\sqrt{nh^{d_A + 4}}}{2}\int u^2 k(u) du \int_\mathcal{X}\int_\mathcal{M}
\left[ \hat{\lambda}_i(a, X_i) - \lambda(a, X_i)\right]
\left[ \frac{\hat{\alpha}_i(a^\prime, M_i, X_i)}{\hat{\alpha}_i(a, M_i, X_i)} - \frac{\alpha_i(a^\prime, M_i, X_i)}{\alpha_i(a, M_i, X_i)}\right] \gamma_i(a, M_i, X_i)
\frac{\partial^2}{\partial a^2}[f(M, a \mid X)]f(X) dm dx\\
+ &o_p(1)\sqrt{nh^{d_A + 4}}\int_\mathcal{X}  \int_\mathcal{M}
\left[ \hat{\lambda}_i(a, X_i) - \lambda(a, X_i)\right]
\left[ \frac{\hat{\alpha}_i(a^\prime, M_i, X_i)}{\hat{\alpha}_i(a, M_i, X_i)} - \frac{ \alpha_i(a^\prime, M_i, X_i)}{\alpha_i(a, M_i, X_i)}\right] \gamma_i(a, M_i, X_i) f(X) dm dx
\end{align*}

Using Assumptions 2, 3 and 4, the above can be written as

\begin{align*}
\sqrt{nh^{d_A}}\int_\mathcal{X}  \int_\mathcal{M}
\left[ \hat{\lambda}_i(a, X_i) - \lambda(a, X_i)\right]
\left[ \frac{\hat{\alpha}_i(a^\prime, M_i, X_i)}{\hat{\alpha}_i(a, M_i, X_i)} - \frac{ \alpha_i(a^\prime, M_i, X_i)}{\alpha_i(a, M_i, X_i)}\right] \gamma_i(a, M_i, X_i)
f(M, a, X) dm dx + o_p(1)
\end{align*}

Now, applying a similar argument to T1, it can be written as

\begin{align*}
\sqrt{nh^{d_A}}\int_\mathcal{X}  \int_\mathcal{M}
\left[ \hat{\lambda}_i(a, X_i) - \lambda(a, X_i)\right]
\left[ \frac{\hat{\alpha}_i(a^\prime, M_i, X_i)}{\hat{\alpha}_i(a, M_i, X_i)} - \frac{ \alpha_i(a^\prime, M_i, X_i)}{\alpha_i(a, M_i, X_i)}\right] \gamma_i(a, M_i, X_i)
f(M, a, X) dm dx + o_p(1)
\end{align*}

Therefore $1 = T1 - T2 = o_p(1)$. 2 and 3 can be similarly bounded following a similar procedure.

Next, we show 4, 5, 6, 8, 11 to be $o_p(1)$. We prove it for 4 and 5, and a similar argument can be used for the rest. Using the Conditional Markov Inequality, Cauchy-Schwartz and Lemma 6.1, to bound 5 it suffices to bound

\begin{align*}
\mathbb{E}\left[\left|\sqrt{\frac{h^{d_A}}{n}} \sum_{i \in \ell} K_h(A_i - a)\lambda(a, X_i)\left[\frac{\hat{\alpha}(a^\prime, M_i, X_i)}{\hat{\alpha}(a, M_i, X_i)} - \frac{\alpha(a^\prime, M_i, X_i)}{\alpha(a, M_i, X_i)} \right]\left[\hat{\gamma}(a, M_i, X_i) - \gamma(a, M_i, X_i) \right] \right| \mid O_c^l \right]\\
\end{align*}

An application of the triangle inequality and the i.i.d assumption gives

\begin{align*}
\leq \sqrt{nh^{d_A}}\mathbb{E}\left[\left|K_h(A_i - a)\lambda(a, X_i)\left[\frac{\hat{\alpha}(a^\prime, M_i, X_i)}{\hat{\alpha}(a, M_i, X_i)} - \frac{\alpha(a^\prime, M_i, X_i)}{\alpha(a, M_i, X_i)} \right]\left[\hat{\gamma}(a, M_i, X_i) - \gamma(a, M_i, X_i) \right] \right| \mid O_c^l \right]\\
\end{align*}

From Assumptions 2 and 3, the above term is broken into

\begin{align*}\tag{CS-1}
\sqrt{nh^{d_A}}\int_{\mathcal{X}}\int_{\mathcal{M}}\lambda(a, X_i)\left|\left[\frac{\hat{\alpha}(a^\prime, M_i, X_i)}{\hat{\alpha}(a, X_i)} - \frac{\alpha(a^\prime, M_i, X_i)}{\alpha(a, M_i, X_i)} \right]\left[\hat{\gamma}(a, M_i, X_i) - \gamma(a, M_i, X_i) \right] \right|f(M, a, X) dm dx\\
\end{align*}

\begin{align*}\tag{CS-2}
\frac{\sqrt{nh^{d_A + 4}}}{2} \int u^2 k(u) du \int_{\mathcal{X}}\int_{\mathcal{M}}\lambda(a, X_i)\left|\left[\frac{\hat{\alpha}(a^\prime, M_i, X_i)}{\hat{\alpha}(a, M_i, X_i)} - \frac{\alpha(a^\prime, M_i, X_i)}{\alpha(a, M_i, X_i)} \right]\left[\hat{\gamma}(a, M_i, X_i) - \gamma(a, M_i, X_i) \right] \right|\left(\frac{\partial^2}{\partial a^2}f(M \mid a, X) f(a \mid X)\right) f(X) dm dx\\
\end{align*}

\begin{align*}\tag{CS-3}
\sqrt{nh^{d_A}}h^2 o_p(1) \int_{\mathcal{X}}\int_{\mathcal{M}}\lambda(a, X_i)\left|\left[\frac{\hat{\alpha}(a^\prime, M_i, X_i)}{\hat{\alpha}(a, M_i, X_i)} - \frac{\alpha(a^\prime, M_i, X_i)}{\alpha(a, M_i, X_i)} \right]\left[\hat{\gamma}(a, M_i, X_i) - \gamma(a, M_i, X_i) \right] \right| f(X) dm dx\\
\end{align*}

CS-3 can be bounded by an application of Holders inequality combined with Assumptions 2, 3 and 4. CS-2 can be bounded similarly. To bound CS-1, note

\begin{align*}
CS1 &\leq \sqrt{nh^{d_A}}|| \lambda(a, X_i)||_{\infty} \int_{\mathcal{X}}\int_{\mathcal{M}}\left|\left[\frac{\hat{\alpha}(a^\prime, M_i, X_i)}{\hat{\alpha}(a, M_i, X_i)} - \frac{\alpha(a^\prime, M_i, X_i)}{\alpha(a, M_i, X_i)} \right]\left[\hat{\gamma}(a, M_i, X_i) - \gamma(a, M_i, X_i) \right] \right|f(M, a, X) dm dx\\
&\leq \sqrt{nh^{d_A}}|| \lambda(a, X_i)||_{\infty} \left(\int_{\mathcal{X}}\int_{\mathcal{M}}\left(\left[\frac{\hat{\alpha}(a^\prime, M_i, X_i)}{\hat{\alpha}(a, M_i, X_i)} - \frac{\alpha(a^\prime, M_i, X_i)}{\alpha(a, M_i, X_i)} \right]\right)^2f(M, a, X) dm dx\right)^{\frac{1}{2}}\\
&\times\left(\int_{\mathcal{X}}\int_{\mathcal{M}} \left(\hat{\gamma}(a, M_i, X_i) - \gamma(a, M_i, X_i) \right)^2f(M, a, X) dm dx\right)^{\frac{1}{2}}
\end{align*}

And the bound follows from Assumption 4.

Similarly, 4 can be bounded by bounding

\begin{align*}
\sqrt{nh^{d_A}} \int_{\mathcal{X}}\int_{\mathcal{M}}\int_{\mathcal{T}}K_h(A - a) \left[\hat{\lambda}(a, X_i) - \lambda(a, X_i) \right]\left[\frac{\hat{\alpha}(a^\prime, M_i, X_i)}{\hat{\alpha}(a, M_i, X_i)} - \frac{\alpha(a^\prime, M_i, X_i)}{\alpha(a, M_i, X_i)} \right]\left[\hat{\gamma}(a, M_i, X_i) - \gamma(a, M_i, X_i) \right]f(M, A, X) da dm dx
\end{align*}

Which can be expanded as

\begin{align*}
&\sqrt{nh^{d_A}} \int_{\mathcal{X}}\int_{\mathcal{M}} \left[\hat{\lambda}(a, X_i) - \lambda(a, X_i) \right]\left[\frac{\hat{\alpha}(a^\prime, M_i, X_i)}{\hat{\alpha}(a, M_i, X_i)} - \frac{\alpha(a^\prime, M_i, X_i)}{\alpha(a, M_i, X_i)} \right]\left[\hat{\gamma}(a, M_i, X_i) - \gamma(a, M_i, X_i) \right]f(M\mid a, X)f(a \mid X) f(X) dm dx\\
+&\sqrt{nh^{d_A + 4}}\int u^2 k(u) du \int_{\mathcal{X}}\int_{\mathcal{M}} \left[\hat{\lambda}(a, X_i) - \lambda(a, X_i) \right]\left[\frac{\hat{\alpha}(a^\prime, M_i, X_i)}{\hat{\alpha}(a, M_i, X_i)} - \frac{\alpha(a^\prime, M_i, X_i)}{\alpha(a, M_i, X_i)} \right]\left[\hat{\gamma}(a, M_i, X_i) - \gamma(a, M_i, X_i) \right]\left(\frac{\partial^2}{\partial a^\prime}f(M\mid a, X)f(a \mid X) \right) f(X) dm dx\\
+&\sqrt{nh^{d_A + 4} } o_p(1) \int_{\mathcal{X}}\int_{\mathcal{M}} \left[\hat{\lambda}(a, X_i) - \lambda(a, X_i) \right]\left[\frac{\hat{\alpha}(a^\prime, M_i, X_i)}{\hat{\alpha}(a, M_i, X_i)} - \frac{\alpha(a^\prime, M_i, X_i)}{\alpha(a, M_i, X_i)} \right]\left[\hat{\gamma}(a, M_i, X_i) - \gamma(a, M_i, X_i) \right]f(X) dm dx
\end{align*}

The first term can be seen to be $o_p(1)$ from Assumption 4. Similarly the second term is $o_p(1)$ from Assumptions 2, 3 and 4. The third term is similarly bounded. Similar approaches can be used to bound 6, 8 and 11.

The left over terms are 7, 9, 10, 12, 13 and 14. The proof for bounding 14 and is given below.

First, add and subtract $\mathbb{E}\left[\sqrt{\frac{h^{d_t}}{n}} \sum_{i \in \ell} \left[\hat{\eta}(a, a^\prime, X_i) - \eta(a, a^\prime, X_i) \right] \mid O_c^l\right]$. Define and note that

\begin{align*}
&\Delta_{il} = \hat{\eta}(a, a', X_i) - \eta(a, a', X_i) - \mathbb{E}\left[ \hat{\eta}(a, a', X_i) - \eta(a, a', X_i) \mid O_c^l\right]\\
\mathbb{E}[&\Delta_{il} \mid Z_c^l] = 0\\
\mathbb{E}[&\Delta_{il}\Delta_{jl} \mid Z_c^l] = 0\\
\mathbb{E}[&\Delta_{il}^2 \mid Z_c^l] = \mathbb{E}\left[\left(\hat{\eta}(a, a', X_i) - \eta(a, a', X_i)\right)^2\right] - \mathbb{E}\left[ \hat{\eta}(a, a', X_i) - \eta(a, a', X_i)\right]^2\\
\mathbb{E}[&\Delta_{il}^2 \mid Z_c^l] \leq \mathbb{E}\left[\left(\hat{\eta}(a, a', X_i) - \eta(a, a', X_i)\right)^2\right] \xrightarrow[]{P} 0\\
\mathbb{E}[&\left(\sqrt{\frac{h^{d_A}}{n}} \sum_{i \in \ell}^n \Delta_{il}^2 \right)^2 \mid O_c^l] \leq h^{d_t}\mathbb{E}[\Delta_{il}^2 \mid Z_c^l] \xrightarrow[]{P} 0
\end{align*}

Hence the terms remaining are: 7, 9, 10, 12, 13. All of these terms are bounded using a similar approach, with an added step to deal with the kernel, that is demonstrated for 7 below.

\begin{align*}
\Delta_{il} = \left\{-K_h(A_i - a)
\lambda_i(a, X_i)
\frac{\alpha_i(a^\prime, M_i, X_i)}{\alpha_i(a, M_i, X_i)}
\left[\hat{\gamma}_i(a, M_i, X_i) - \gamma_i(a, M_i, X_i) \right]
\right\}\\
+ \mathbb{E}\left[-K_h(A_i - a)
\lambda_i(a, X_i)
\frac{\alpha_i(a^\prime, M_i, X_i)}{\alpha_i(a, M_i, X_i)}
\left[\hat{\gamma}_i(a, M_i, X_i) - \gamma_i(a, M_i, X_i) \right]
\mid O_c^l \right]
\end{align*}

\begin{align*}
\mathbb{E}\left[\Delta_{il} \mid Z_c^l\right] &= 0\\
\mathbb{E}\left[\Delta_{il}\Delta_{jl} \mid Z_c^l\right] &= 0\\
\mathbb{E}\left[\Delta_{il}^2 \mid Z_c^l \right] &\leq \int_{\mathcal{X}}\int_{\mathcal{M}}\int_{\mathcal{A}} K_h^2(A_i - a)\lambda^2(a, X_i)\frac{\alpha^2(a, M_i, X_i)}{\alpha^2(a^\prime, M_i, X_i)}\left[\hat{\gamma}(a, M_i, X_i) - \gamma(a, M_i, X_i) \right]^2f(M, A, X) dx dm da
\end{align*}

Note that $h^{d_A}\mathbb{E}\left[K_h^2(A_i - a)\lambda^2(a, X_i)\frac{\alpha^2(a, M_i, X_i)}{\alpha^2(a^\prime, M_i, X_i)}\left[\hat{\gamma}(a, M_i, X_i) - \gamma(a, M_i, X_i) \right]^2\right] \xrightarrow[]{P} 0$, which follows from Assumption 4.

Bounding each of the terms above introduced additional terms, that we group together as "robust" terms below.

\begin{align*}\tag{R1}
- &\mathbb{E} \left[\sqrt{\frac{h^{d_A}}{n}}\sum_{i \in \ell}\left\{K_h(A_i - a)\lambda_i(a, X_i)\frac{\alpha_i(a^\prime, M_i, X_i)}{\alpha_i(a, M_i, X_i)}\left[\hat{\gamma}_i(a, M_i, X_i) - \gamma_i(a, M_i, X_i) \right]\right\}\right]
\end{align*}

\begin{align*}\tag{R2}
+ &\mathbb{E}\left[\sqrt{\frac{h^{d_A}}{n}} \sum_{i \in \ell} \left\{K_h(A_i - a^\prime) \left[\hat{\lambda}_i(a^\prime, M_i, X_i) - \lambda(a^\prime, M_i, X_i) \right]\gamma_i(a, M_i, X_i)\right\}  \right]
\end{align*}

\begin{align*}\tag{R3}
+ &\mathbb{E}\left[\sqrt{\frac{h^{d_A}}{n}} \sum_{i \in \ell} \left\{K_h(A_i - a^\prime) \lambda_i(a^\prime, X_i)\left[\hat{\gamma}_i(a, M_i, X_i) - \gamma_i(a, M_i, X_i) \right] \right\}  \right]
\end{align*}

\begin{align*}\tag{R4}
- &\mathbb{E}\left[\sqrt{\frac{h^{d_A}}{n}} \sum_{i \in \ell}^n \left\{K_h(A_i - a) \left[\hat{\lambda}_i(a^\prime, X_i) - \lambda(a^\prime, X_i) \right]\eta(a, a', X_i)\right\}\right]
\end{align*}

\begin{align*}\tag{R5}
- & \mathbb{E}\left[\sqrt{\frac{h^{d_A}}{n}}\sum_{i \in \ell} \left\{K_h(A_i - a^\prime) \lambda(a^\prime, X_i)\left[\hat{\eta}_i(a, a^\prime, X_i) - \eta(a, a^\prime, X_i) \right]\right\}\right]\\
\end{align*}

\begin{align*}\tag{R6}
+ & \mathbb{E}\left[\sqrt{\frac{h^{d_A}}{n}}\sum_{i \in \ell} \left[\hat{\eta}_i(a, a^\prime, X_i) - \eta_i(a,a^\prime, X_i) \right] \right]
\end{align*}

To bound all of these terms, we show that R2 $+$ R4 $\xrightarrow[]{P} 0$, R5 $+$ R6 $\xrightarrow[]{P} 0$, and R1 $+$ R3 $\xrightarrow[]{P} 0$. The bound for R5 $+$ R6 is demonstrated, and the rest follow a similar argument.

\begin{align*}\tag{R5 $+$ R6}
&\sqrt{\frac{h^{d_A}}{n}}\sum_{i \in \ell}
\mathbb{E}\left[(1 - K_h(A_i - a^\prime) \lambda(a^\prime, X_i))\left[\hat{\eta}_i(a, a^\prime, X_i) - \eta(a, a^\prime, X_i) \right]\right]
\end{align*}

\begin{align*}
&\leq \sqrt{nh^{d_A}}\mathbb{E}\left[(1 - K_h(A_i - a^\prime) \lambda(a^\prime, X_i))\left[\hat{\eta}_i(a, a^\prime, X_i) - \eta(a, a^\prime, X_i) \right]\right]\\
& = \sqrt{nh^{d_A}}\mathbb{E}\left[\mathbb{E}\left[(f(a^\prime \mid X) - K_h(A_i - a^\prime)) \mid X \right] \lambda(a^\prime, X_i)\left[\hat{\eta}_i(a, a^\prime, X_i) - \eta(a, a^\prime, X_i) \right]\right]\\
\end{align*}

The inner expectation $\mathbb{E}\left[f(a^\prime \mid X) - K_h(A_i - a^\prime) \mid X \right]$ can be written as

\begin{align*}
& f(a^\prime \mid X) - \int_{\mathcal{A}}K_h(A_i - a^\prime))f(A \mid X) dA\\
&= f(a^\prime \mid X)  -  f(a^\prime \mid X) + \frac{h^2}{2}\int u^2k(u) du\frac{\partial^2}{\partial a^2} f(a^\prime \mid X) + O(h^3)
\end{align*}

Plugging back into original integral

\begin{align*}
\sqrt{nh^{d_A}}\mathbb{E}\left[\frac{h^2}{2}\int u^2k(u) du\left(\frac{\partial^2}{\partial a^2} f(a^\prime \mid X)\right)\lambda(a^\prime, X_i)\left[\hat{\eta}_i(a, a^\prime, X_i) - \eta(a, a^\prime, X_i) \right] + O(h^3)\right]
\end{align*}

This can be bounded as $o_p(1)$ using Assumption 2, 3 and 4.

Next, for bias $B(a,a')$ and variance $V(a,a')$, given

\begin{align*}
&\sqrt{nh^{d_A}} (\hat{\psi}^{TR}(a,a')- \psi_0(a,a') )\\
=& \sqrt{\frac{h^{d_A}}{n}}\sum^L_{l=1}\sum_{i\in I_\ell} \Big\{ m(O_i;\alpha,\lambda,\gamma,\psi_0(a,a'))\Big\}+o_p(1)
\end{align*}

\begin{align*}
m(O_i;\alpha,\lambda,\gamma,\psi_0(a,a')) &= \frac{K_h(A_i - a)f(M_i \mid A = a', X_i)}{f(M_i \mid A = a, X_i)f(a \mid X_i)}\{Y_i - \mathbb{E}[Y \mid X_i, M_i , A = a]\}\\
&+ \frac{K_h(A_i - a')}{f(a' \mid X_i)}\{\mathbb{E}[Y \mid X_i, M_i, A = a] - \eta(a, a', X_i)\} + \eta(a, a', X_i) - \psi_0(a,a')
\end{align*}

we focus on

$$\frac{K_h(A - a)f(M \mid A = a', X)}{f(M \mid A = a, X)f(a \mid X)}\{Y - \mathbb{E}[Y \mid X, M , A = a]\}+ \frac{K_h(A - a')}{f(a' \mid X)}\{\mathbb{E}[Y \mid X, M, A = a] - \eta(a, a', X)\} ,$$

where $\eta(a, a', X) = \int \mathbb{E}[Y \mid X, M = m, A = a]f(m \mid A = a', X) dm = \mathbb{E}\{\mathbb{E}[Y \mid X, A = a]|A = a', X\}$.

Expectation Part 1

$$\frac{K_h(A - a)f(M \mid A = a', X)}{f(M \mid A = a, X)f(a \mid X)}\{Y - \mathbb{E}[Y \mid X, M , A = a]\}$$

From $\mathbb{E}\{\mathbb{E}[\gamma(X,M,A)|X,M]\} = \mathbb{E}\{\mathbb{E}[Y|X,M]\}$ and $\gamma(X,M,A) = \mathbb{E}(Y|X,M,A)$, expectation of the first term

\begin{align*}
&\mathbb{E}\bigg[\frac{K_h(A - a)f(M \mid A = a', X)}{f(M \mid A = a, X)f(a \mid X)}\{Y - \mathbb{E}[Y \mid X, M , A = a]\}\bigg]   \\
=& \mathbb{E}\Bigg\{\mathbb{E}\bigg[\frac{K_h(A - a)f(M \mid A = a', X)}{f(M \mid A = a, X)f(a \mid X)}\{Y - \mathbb{E}[Y \mid X, M , A = a]\}\bigg| X,M\bigg]\Bigg\}\\
=& \mathbb{E}\Bigg\{\frac{f(M \mid A = a', X)}{f(M \mid A = a, X)f(a \mid X)} \mathbb{E}\bigg[K_h(A - a) (\gamma(X,M,A) - \gamma(X,M,a))\bigg| X,M\bigg]\Bigg\}.
\end{align*}

The inner product further expands as follows,

\begin{align*}
&\mathbb{E}\bigg[K_h(A - a) (\gamma(X,M,A) - \gamma(X,M,a))\bigg| X,M\bigg]\\
=& \int_\mathcal{A} K_h(A - a) (\gamma(X,M,A) - \gamma(X,M,a)) f(A|X,M) dA\\
=& \int_\mathcal{A} \bigg[\prod^{d_A}_{j=1}\frac{1}{h}k\Big(\frac{A_j-a}{h}\Big)\bigg] (\gamma(X,M,A) - \gamma(X,M,a)) f(A|X,M) dA\\
=& \int k(u) (\gamma(X,M,a+uh) - \gamma(X,M,a)) f(a+uh|X,M) du\\
=& \int k(u_1)\cdots k(u_{d_A}) \bigg(\sum^{d_A}_{j=1} u_j h\partial_{a_j} \gamma(X,M,a) +\frac{u^2_jh^2}{2} \partial^2_{a_j} \gamma(X,M,a) \bigg)\\
& \times \bigg( f(a|X,M)+\sum^{d_A}_{j=1} u_j h\partial_{a_j} f(a|X,M) +\frac{u^2_j h^2}{2} \partial^2_{a_j} f(a|X,M)\bigg)du_1\cdots du_{d_A} +O(h^3)\\
=& h^2 \int u^2 k(u)du \bigg(\sum^{d_A}_{j=1} \partial_{a_j} \gamma(X,M,a) \partial_{a_j} f(a|X,M) + \frac{1}{2} [\partial^2_{a_j} \gamma(X,M,a)]f(a|X,M) \bigg) + O(h^3)
\end{align*}

for all $X,M$ in respective range.

\begin{align*}
&\mathbb{E}\Bigg\{\frac{f(M \mid A = a', X)}{f(M \mid A = a, X)f(a \mid X)} \mathbb{E}\bigg[K_h(A - a) (\gamma(X,M,A) - \gamma(X,M,a))\bigg| X,M\bigg]\Bigg\} \\
=& h^2 \int u^2 k(u)du \\
&\times \mathbb{E}\bigg[ \frac{f(M \mid A = a', X)}{f(M \mid A = a, X)}\bigg(
\partial_a \gamma(X,M,a) \frac{\partial_a f(a|X,M)}{f(a \mid X)} + \frac{1}{2} \frac{f(a\mid X,M)}{f(a\mid X)}  \partial^2_a \gamma(X,M,a)
\bigg)\bigg] + O(h^3)
\end{align*}

Expectation Part 2

$$\frac{K_h(A - a')}{f(a' \mid X)}\{\mathbb{E}[Y \mid X, M, A = a] - \eta(a, a', X)\}$$

\begin{align*}
&\mathbb{E}\bigg[\frac{K_h(A - a')}{f(a' \mid X)}\{\mathbb{E}[Y \mid X, M, A = a] - \eta(a, a', X)\}\bigg]\\
=&\mathbb{E}\bigg\{\mathbb{E}\bigg[\frac{K_h(A - a')}{f(a' \mid X)}\{\mathbb{E}[Y \mid X, M, A = a] - \eta(a, a', X)\}\bigg| X,M\bigg]\bigg\} \\
=&\mathbb{E}\bigg\{\frac{1}{f(a' \mid X)}\mathbb{E}\bigg[K_h(A - a')\{\mathbb{E}[Y \mid X, M, A = a] - \eta(a, a', X)\}\bigg| X,M\bigg]\bigg\}\\
=&\mathbb{E}\bigg\{\frac{\mathbb{E}[Y \mid X, M, A = a] - \eta(a, a', X)}{f(a' \mid X)}\mathbb{E}[K_h(A - a')\bigg| X,M]\bigg\}
\end{align*}

The inner expectation becomes

\begin{align*}
&\mathbb{E}\bigg[K_h(A - a')\bigg| X,M\bigg]\\
=& \int_\mathcal{A} \bigg[\prod^{d_A}_{j=1}\frac{1}{h}k\Big(\frac{A_j-a'}{h}\Big)\bigg]  f(A|X,M) dA \\
=& \int k(u_1)\cdots k(u_{d_A}) \bigg( f(a'|X,M)+ \sum^{d_A}_{j=1} u_jh\partial_{a_j} f(a'|X,M) +\frac{u^2_jh^2}{2} \partial^2_{a_j} f(a'|X,M)\bigg)du_1\cdots du_{d_A} +O(h^3)\\
=&f(a'|X,M) + \frac{1}{2}h^2\int u^2k(u)du \sum^{d_A}_{j=1}\partial^2_{a_j} f(a'|X,M) + O(h^3)
\end{align*}

The expectation of part 2

\begin{align*}
&\mathbb{E}\Bigg\{ \frac{1}{f(a'|X)}\{\mathbb{E}[Y \mid X, M, A = a] - \eta(a, a', X)\}\\ &\times \bigg(f(a'|X,M) + \frac{1}{2}h^2\int u^2k(u)du \partial^2_a f(a'|X,M) \bigg)\Bigg\} + O(h^3)\\
=& \mathbb{E}\Bigg\{ \{\mathbb{E}[Y \mid X, M, A = a] - \eta(a, a', X)\}\\ &\times \bigg(\frac{f(a'|X,M)}{f(a'|X)} + \frac{1}{2}h^2\int u^2k(u)du \frac{\partial^2_a f(a'|X,M)}{f(a'|X)} \bigg)\Bigg\} + O(h^3)\\
=& h^2\int u^2k(u)du\mathbb{E}\Bigg\{ \{\mathbb{E}[Y \mid X, M, A = a] - \eta(a, a', X)\}\frac{1}{2}\frac{\partial^2_a f(a'|X,M)}{f(a'|X)}\Bigg\}+ O(h^3)\\
\end{align*}

from having the first term in this expectation equal to zero,

\begin{align*}
& \mathbb{E}\Bigg\{ \{\mathbb{E}[Y \mid X, M, A = a] - \eta(a, a', X)\}\frac{f(a'|X,M)}{f(a'|X)} \Bigg\} \\
=& \int\int \{\mathbb{E}[Y \mid X, M, A = a] - \eta(a, a', X)\}\frac{f(a'|X,M)}{f(a'|X)} f(M,X)dMdX \\
=& \int\int \{\mathbb{E}[Y \mid X, M, A = a] - \eta(a, a', X)\}\frac{f(A=a',X,M)}{f(A = a', X)} f(X)dMdX \\
=& \int\int \{\mathbb{E}[Y \mid X, M, A = a] - \eta(a, a', X)\}f(M|A=a',X) dM f(X)dX \\
=& \int \{\eta(a, a', X)-\eta(a, a', X)\} f(X)dX=0
\end{align*}

\begin{align*}
B(a,a') =& \\
& h^2 \int u^2 k(u)du \\
&\times \mathbb{E}\Bigg\{ \frac{f(M \mid A = a', X)}{f(M \mid A = a, X)}\bigg(
\partial_a \gamma(X,M,a) \frac{\partial_a f(a|X,M)}{f(a \mid X)} + \frac{1}{2} \frac{f(a\mid X,M)}{f(a\mid X)}  \partial^2_a \gamma(X,M,a)
\bigg) \\
&+ \{\mathbb{E}[Y \mid X, M, A = a] - \eta(a, a', X)\}  \frac{1}{2} \frac{\partial^2_a f(a'|X,M)}{f(a'|X)} \Bigg\}+O(h^3)
\end{align*}

\begin{align*}
&h^{d^A}\times var\Bigg\{\frac{K_h(A - a)f(M \mid A = a', X)}{f(M \mid A = a, X)f(a \mid X)}\{Y - \mathbb{E}[Y \mid X, M , A = a]\} \\
& \qquad + \frac{K_h(A - a')}{f(a' \mid X)}\{\mathbb{E}[Y \mid X, M, A = a] - \eta(a, a', X)\}\\
& \qquad +  \eta(a, a', X) - \theta_0\Bigg\}
\end{align*}

To start with, we look at the second order expectation

\begin{align*}
&\mathbb{E}\Bigg\{\bigg[\frac{K_h(A - a)f(M \mid A = a', X)}{f(M \mid A = a, X)f(a \mid X)}\{Y - \mathbb{E}[Y \mid X, M , A = a]\} \\
& + \frac{K_h(A - a')}{f(a' \mid X)}\{\mathbb{E}[Y \mid X, M, A = a] - \eta(a, a', X)\} \bigg]^2\Bigg\} \\
=& \mathbb{E}\Bigg\{\bigg[\frac{K_h(A - a)f(M \mid A = a', X)}{f(M \mid A = a, X)f(a \mid X)}\{Y - \mathbb{E}[Y \mid X, M , A = a]\}  \bigg]^2\Bigg\} \\
&+ \mathbb{E}\Bigg\{\bigg[\frac{K_h(A - a')}{f(a' \mid X)}\{\mathbb{E}[Y \mid X, M, A = a] - \eta(a, a', X)\} \bigg]^2\Bigg\} \\
&+ 2\mathbb{E}\Bigg\{\bigg[\frac{K_h(A - a)f(M \mid A = a', X)}{f(M \mid A = a, X)f(a \mid X)}\{Y - \mathbb{E}[Y \mid X, M , A = a]\}  \bigg]\\
&\qquad \times \bigg[\frac{K_h(A - a')}{f(a' \mid X)}\{\mathbb{E}[Y \mid X, M, A = a] - \eta(a, a', X)\} \bigg]\Bigg\}
\end{align*}

Variance Part 1

\begin{align*}
&\mathbb{E}\Bigg\{\bigg[\frac{K_h(A - a)f(M \mid A = a', X)}{f(M \mid A = a, X)f(a \mid X)}\{Y - \mathbb{E}[Y \mid X, M , A = a]\}  \bigg]^2\Bigg\}\\
=&\mathbb{E}\Bigg\{\mathbb{E}\bigg\{\bigg[\frac{K_h(A - a)f(M \mid A = a', X)}{f(M \mid A = a, X)f(a \mid X)}\{Y - \mathbb{E}[Y \mid X, M , A = a]\}  \bigg]^2\bigg|X,M\bigg\}\Bigg\}\\
=&\mathbb{E}\Bigg\{\frac{f(M \mid A = a', X)^2}{f(M \mid A = a, X)^2f(a \mid X)^2}\mathbb{E}\bigg\{K_h(A - a)^2(Y - \mathbb{E}[Y \mid X, M , A = a])^2\bigg|X,M\bigg\}\Bigg\}\\
=&\mathbb{E}\Bigg\{\frac{f(M \mid A = a', X)^2}{f(M \mid A = a, X)^2f(a \mid X)^2}\\
&\times \mathbb{E}\bigg\{K_h(A - a)^2\mathbb{E}\{(Y - \mathbb{E}[Y \mid X, M , A = a])^2|X,M,A\}\bigg|X,M\bigg\}\Bigg\}\\
=&\mathbb{E}\Bigg\{\frac{f(M \mid A = a', X)^2}{f(M \mid A = a, X)^2f(a \mid X)^2}\\
&\times \mathbb{E}\bigg\{K_h(A - a)^2 \bigg[var(Y|X,M,A) + \gamma(X,M,A)^2 -2\gamma(X,M,A)\gamma(X,M,a)+\gamma(X,M,a)^2\bigg]
\bigg|X,M\bigg\}\Bigg\}\\
=&\mathbb{E}\Bigg\{\frac{f(M \mid A = a', X)^2}{f(M \mid A = a, X)^2f(a \mid X)^2}\\
&\times \mathbb{E}\bigg\{K_h(A - a)^2 \bigg[var(Y|X,M,A) + (\gamma(X,M,A)-\gamma(X,M,a))^2\bigg]
\bigg|X,M\bigg\}\Bigg\}
\end{align*}

The inner expectation

\begin{align*}
&\mathbb{E}\bigg\{K_h(A - a)^2 \bigg[var(Y|X,M,A) + (\gamma(X,M,A)-\gamma(X,M,a))^2\bigg]
\bigg|X,M\bigg\}\\
= &\int \bigg[\prod^{d_A}_{j=1}\frac{1}{h^2}k\Big(\frac{A_j-a}{h}\Big)^2\bigg]  \bigg[var(Y|X,M,A) + (\gamma(X,M,A)-\gamma(X,M,a))^2\bigg]
f(A|X,M) dA\\
= & \frac{1}{h^{d_A}} \int k(u)^2 \\
&\times \bigg[var(Y|X,M,a+uh) + (\gamma(X,M,a+uh)-\gamma(X,M,a))^2\bigg]
f(a+uh|X,M) du\\
= & \frac{1}{h^{d_A}} \int k(u_1)^2\cdots k(u_{d_A})^2 \\
&\times \bigg[var(Y|X,M,a)+\sum^{d_A}_{j=1}u_jh\partial_{a_j} var(Y|X,M,a) + \frac{u^2_jh^2}{2}\partial^2_{a_j} var(Y|X,M,a) +O(h^3)\\
&\qquad + \bigg( \sum^{d_A}_{j=1} u_jh\partial_{a_j} \gamma(X,M,a)+ \frac{u^2_jh^2}{2}\partial^2_{a_j} \gamma(X,M,a)+O(h^3)\bigg)^2\bigg]\\
&\times \bigg(f(a|X,M)+\sum^{d_A}_{j=1}u_jh\partial_{a_j} f(a|X,M) +  \frac{u^2_jh^2}{2}\partial^2_{a_j} f(a|X,M) + O(h^3) \bigg)du_1\cdots du_{d_A}\\
=& \frac{1}{h^{d_A}}\Bigg\{\int k(u)^2 du \times  var(Y|X,M,a)f(a|X,M)    \\
&+ h^2\int u^2k(u)^2du\\ &\qquad\times\bigg\{\bigg[\frac{1}{2}\partial^2_a var(Y|X,M,a) + (\partial_a \gamma(X,M,a))^2 \bigg]f(a|X,M)+ \partial_a var(Y|X,M,a)\partial_a f(a|X,M)\\
&\qquad +\frac{1}{2}var(Y|X,M,a) \partial^2_a f(a|X,M)\bigg\}+O(h^4)\Bigg\}
\end{align*}

Hence, the part 1 of variance

\begin{align*}
&\mathbb{E}\Bigg\{\bigg[\frac{K_h(A - a)f(M \mid a', X)}{f(M \mid a, X)f(a \mid X)}\{Y - \mathbb{E}[Y \mid X, M , a]\}  \bigg]^2\Bigg\}\\
=&\mathbb{E}\Bigg\{\frac{f(M \mid a', X)^2}{f(M \mid a, X)^2f(a \mid X)^2}\\
&\times \Bigg[\frac{1}{h^{d_A}}\int k(u)^2 du \times  var(Y|X,M,a)f(a|X,M)    \\
&\qquad + h^{2-d_A}\int u^2k(u)^2du\times\bigg(\Big[\frac{1}{2}\partial^2_a var(Y|X,M,a) + (\partial_a \gamma(X,M,a))^2 \Big]f(a|X,M)\\
&\qquad+ \partial_a var(Y|X,M,a)\partial_a f(a|X,M)+\frac{1}{2}var(Y|X,M,a) \partial^2_a f(a|X,M)\bigg)\Bigg] \Bigg\}+O(h^{4-d_A})\\
=& \frac{1}{h^{d_A}}\int k(u)^2 du \times \mathbb{E}\Bigg\{\frac{f(M \mid a', X)^2 f(a|X,M)}{f(M \mid a, X)^2f(a \mid X)^2} var(Y|X,M,a)\Bigg\}\\
&+ h^{2-d_A}\int k(u)^2u^2du\times \mathbb{E}\Bigg\{\frac{f(M \mid a', X)^2 f(a|X,M)}{f(M \mid a, X)^2f(a \mid X)^2} \Big[\frac{1}{2}\partial^2_a var(Y|X,M,a) + (\partial_a \gamma(X,M,a))^2 \Big]\\
&\qquad + \frac{f(M \mid a', X)^2 }{f(M \mid a, X)^2f(a \mid X)^2} \Big[\partial_a var(Y|X,M,a)\partial_a f(a|X,M)+\frac{1}{2}var(Y|X,M,a) \partial^2_a f(a|X,M)\Big]\Bigg\}\\
& + O(h^{4-d_A})
\end{align*}

Variance Part 2

\begin{align*}
&\mathbb{E}\Bigg\{\bigg[\frac{K_h(A - a')}{f(a' \mid X)}\Big(\mathbb{E}(Y \mid X, M, A = a) - \eta(a, a', X)\Big) \bigg]^2\Bigg\}\\
=&\mathbb{E}\Bigg\{\frac{1}{f(a' \mid X)^2}\mathbb{E}\bigg[K_h(A - a')^2\Big(\gamma(X, M, a) - \eta(a, a', X)\Big)^2\Big|X\bigg]\Bigg\}\\
\end{align*}

The inner expectation

\begin{align*}
&\mathbb{E}\bigg[K_h(A - a')^2\Big(\gamma(X, M, a) - \eta(a, a', X)\Big)^2\Big|X\bigg]\\
=& \int\int \bigg[\prod^{d_A}_{j=1}\frac{1}{h^2}k\Big(\frac{A_j-a'}{h}\Big)^2\bigg]  \Big(\gamma(X, M, a) - \eta(a, a', X)\Big)^2 f(A|X)f(M|A,X)dAdM \\
=& \frac{1}{h^{d_A}}\int\int k(u)^2\Big(\gamma(X, M, a) - \eta(a, a', X)\Big)^2 f(a'+uh|X)f(M|a'+uh,X)dudM\\
=& \frac{1}{h^{d_A}}\int\int k(u_1)^2\cdots k(u_{d_A})^2\Big(\gamma(X, M, a) - \eta(a, a', X)\Big)^2\\
&\times \Big(f(a'|X)+\sum^{d_A}_{j=1}u_j h\partial_{a_j} f(a'|X)+\frac{u^2_j h^2}{2}\partial^2_{a_j} f(a'|X)+O(h^3)\Big)\\
&\times \Big(f(M|a',X)+\sum^{d_A}_{j=1}u_j h\partial_{a_j} f(M|a',X)+\frac{u^2_j h^2}{2}\partial^2_{a_j} f(M|a',X) + O(h^3)\Big)du_1\cdots du_{d_A}dM\\
=& \frac{1}{h^{d_A}} \int k(u)^2 du \times var[E(Y|X,M,a)|X,a']f(a'|X)\\
&+ h^{2-d_A} \int k(u)^2u^2du \times \bigg\{\frac{1}{2}var[E(Y|X,M,a)|X,a']\partial^2_a f(a'|X)\\
&+ \int \Big(\gamma(X, M, a) - \eta(a, a', X)\Big)^2\times \Big[\frac{1}{2}f(a'|X)\partial^2_a f(M|X,a') + \partial_a f(a'|X) \partial_a f(M|X,a') \Big]dM \bigg\}\\
&+O(h^{4- d_A})
\end{align*}

the last equation is from

\begin{align*}
var[E(Y|X,M,a)|X,a']=& \mathbb{E}\bigg\{\Big[E(Y|X,M,a)-\eta(a,a',X)\Big]^2|X,a'\bigg\}\\
=& \int \Big[E(Y|X,M,a)-\eta(a,a',X)\Big]^2 f(M|X,a')dM.
\end{align*}

Hence, the part 2 of variance

\begin{align*}
&\mathbb{E}\Bigg\{\frac{1}{f(a' \mid X)^2}\mathbb{E}\bigg[K_h(A - a')^2\Big(\gamma(X, M, a) - \eta(a, a', X)\Big)^2\Big|X\bigg]\Bigg\}\\
=& \frac{1}{h^{d_A}} \int k(u)^2 du \times \mathbb{E}\bigg\{\frac{1}{f(a'|X)}var[E(Y|X,M,a)|X,a']\bigg\}\\
&+ h^{2-d_A} \int k(u)^2u^2du \times \frac{1}{2} \mathbb{E}\bigg\{\frac{\partial^2_a f(a'|X)}{f(a'|X)^2}var[E(Y|X,M,a)|X,a']\bigg\}\\
&+ h^{2-d_A} \int k(u)^2u^2du\times \\
&\qquad \mathbb{E}\bigg\{ \Big(\gamma(X, M, a) - \eta(a, a', X)\Big)^2\times \Big[\frac{\partial^2_a f(M|X,a')}{2f(a'|X)f(M|X,A)} + \frac{\partial_a f(a'|X) \partial_a f(M|X,a')}{f(a'|X)^2 f(M|X,A)} \Big] \bigg\}\\
\end{align*}

Variance Part 3

\begin{align*}
&\mathbb{E}\Bigg\{\bigg[\frac{K_h(A - a)f(M \mid A = a', X)}{f(M \mid A = a, X)f(a \mid X)}\{Y - \mathbb{E}[Y \mid X, M , A = a]\}  \bigg]\\
&\qquad \times \bigg[\frac{K_h(A - a')}{f(a' \mid X)}\{\mathbb{E}[Y \mid X, M, A = a] - \eta(a, a', X)\} \bigg]\Bigg\}\\
\Big[\gamma(X,M,a)-\eta(a,a',X)\Big]\\
& \qquad\times\mathbb{E}\Big\{K_h(A-a)K_h(A-a')\Big[Y-\gamma(X,M,a)\Big]\Big|X,M\Big\}\Bigg\}\\
\Big[\gamma(X,M,a)-\eta(a,a',X)\Big]\\
& \qquad\times\mathbb{E}\Big\{K_h(A-a)K_h(A-a')\Big[\gamma(X,M,A)-\gamma(X,M,a)\Big]\Big|X,M\Big\}\Bigg\}
\end{align*}

The inner expectation

\begin{align*}
=& \int \bigg[\prod^{d_A}_{j=1}\frac{1}{h^2}k\Big(\frac{A_j-a}{h}\Big)k\Big(\frac{A_j-a'}{h}\Big)\bigg]
\Big[\gamma(X,M,A)-\gamma(X,M,a)\Big] f(A|X,M)dA\\
=& \int \frac{1}{h^{d_A}} k(u_1)\cdots k(u_{d_A}) k(u_1+\frac{a-a'}{h})\cdots k(u_{d_A}+\frac{a-a'}{h})\\  &\times \Big[\sum^{d_A}_{j=1} u_j h\partial_{a_j} \gamma(X,M,a)+\frac{u^2_j h^2}{2}\partial^2_{a_j}\gamma(X,M,a)+\frac{u^3_jh^3}{6}\partial^3_{a_j}\gamma(X,M,a)+O(h^4)\Big]\\
&\times \Big[f(a|X,M) +\sum^{d_A}_{j=1} u_jh\partial_{a_j} f(a|X,M)+\frac{u^2_jh^2}{2}\partial^2_{a_j} f(a|X,M) + O(h^3)\Big]du_1\cdots du_{d_A}\\
=& h^{1-d_A}\int k(u) k(u+\frac{a-a'}{h})udu \times  f(a|X,M)\partial_a \gamma(X,M,a)\\
& + h^{2-d_A}\int k(u) k(u+\frac{a-a'}{h})u^2du \times \Big[\frac{1}{2}f(a|X,M)\partial^2_a \gamma(X,M,a)+\partial_a f(a|X,M)\partial_a \gamma(X,M,a) \Big]\\
& + h^{3-d_A}\int k(u) k(u+\frac{a-a'}{h})u^3du \times\\ &\qquad \Big[\frac{1}{6}f(a|X,M)\partial^3_a \gamma(X,M,a)+\frac{1}{2}\partial_a f(a|X,M)\partial^2_a \gamma(X,M,a)+\frac{1}{2}\partial^2_a f(a|X,M)\partial_a \gamma(X,M,a) \Big]\\
\end{align*}

Hence, the part 3 of variance

\begin{align*}
\Big[\gamma(X,M,a)-\eta(a,a',X)\Big]\\
& \qquad\times\mathbb{E}\Big\{K_h(A-a)K_h(A-a')\Big[\gamma(X,M,A)-\gamma(X,M,a)\Big]\Big|X,M\Big\}\Bigg\}\\
\Big[\gamma(X,M,a)-\eta(a,a',X)\Big]\times\\
&\qquad \bigg[ h^{1-d_A}\int k(u)k(u+\frac{a-a'}{h})udu \times  f(a|X,M)\partial_a \gamma(X,M,a)\\
&\qquad + h^{2-d_A}\int k(u) k(u+\frac{a-a'}{h})u^2du \times \Big[\frac{1}{2}f(a|X,M)\partial^2_a \gamma(X,M,a)+\partial_a f(a|X,M)\partial_a \gamma(X,M,a) \Big]\\
&\qquad + h^{3-d_A}\int k(u) k(u+\frac{a-a'}{h})u^3du \times\\ &\qquad \Big[\frac{1}{6}f(a|X,M)\partial^3_a \gamma(X,M,a)+\frac{1}{2}\partial_a f(a|X,M)\partial^2_a \gamma(X,M,a)+\frac{1}{2}\partial^2_a f(a|X,M)\partial_a \gamma(X,M,a) \Big]\bigg] \Bigg\}\\
&+O(h^{4-d_A})  \\
=& h^{1-d_A}\int k(u) k(u+\frac{a-a'}{h})udu \times \mathbb{E}\bigg\{\frac{f(a'|M,X)}{f(a'|X)^2}\Big[\gamma(X,M,a)-\eta(a,a',X)\Big]\partial_a \gamma(X,M,a)\bigg\}\\
&+ h^{2-d_A}\int k(u) k(u+\frac{a-a'}{h})u^2du\\
&\qquad \times \mathbb{E}\bigg\{\frac{f(a'|M,X)}{f(a'|X)^2}\Big[\gamma(X,M,a)-\eta(a,a',X)\Big]\Big[\frac{1}{2}\partial^2_a \gamma(X,M,a) +\frac{\partial_a f(a|X,M)\partial_a \gamma(X,M,a)}{f(a|M,X)} \Big]\bigg\}\\
&+ h^{3-d_A}\int k(u) k(u+\frac{a-a'}{h})u^3du \times \mathbb{E}\bigg\{\frac{f(a'|M,X)}{f(a'|X)^2}\Big[\gamma(X,M,a)-\eta(a,a',X)\Big]\\
&\qquad \times\Big[\frac{1}{6}\partial^3_a \gamma(X,M,a)+\frac{\partial_a f(a|X,M)\partial^2_a \gamma(X,M,a)}{2f(a|M,X)}+\frac{\partial^2_a f(a|X,M)\partial_a \gamma(X,M,a)}{2f(a|M,X)} \Big]\bigg\}\\
&+O(h^{4-d_A})  \\
\end{align*}

Summarizing the three parts, the variance can be written as

\begin{align*}
V(a,a')=&h^{d_A}\times \bigg\{ \frac{1}{h^{d_A}}\int k(u)^2du \times E_1 + h^{2-d_A} \int k(u)^2 u^2 du \times E_2 \\
&+ h^{1-d_A} \int k(u) k(u+\frac{a-a'}{h})udu \times E_3 + h^{2-d_A}\int k(u) k(u+\frac{a-a'}{h})u^2du \times E_4 \\
& + h^{3-d_A}\int k(u) k(u+\frac{a-a'}{h})u^3du\times E_5 + O(h^{4-d_A})\bigg\}\\
=&\int k(u)^2du \times E_1 + h^{2} \int k(u)^2 u^2 du \times E_2 \\
&+ h \int k(u) k(u+\frac{a-a'}{h})udu \times E_3 + h^2\int k(u) k(u+\frac{a-a'}{h})u^2du \times E_4 \\
& + h^3\int k(u) k(u+\frac{a-a'}{h})u^3du\times E_5 + O(h^4)
\end{align*}

\begin{align*}
E_1 = & \mathbb{E}\Bigg\{\frac{f(M \mid a', X)^2 f(a|X,M)}{f(M \mid a, X)^2f(a \mid X)^2} var(Y|X,M,a)+ \frac{1}{f(a'|X)}var[E(Y|X,M,a)|X,a']\Bigg\}\\
E_2 = & \mathbb{E}\Bigg\{\frac{f(M \mid a', X)^2 f(a|X,M)}{f(M \mid a, X)^2f(a \mid X)^2} \Big[\frac{1}{2}\partial^2_a var(Y|X,M,a) + (\partial_a \gamma(X,M,a))^2 \Big]\\
& + \frac{f(M \mid a', X)^2 }{f(M \mid a, X)^2f(a \mid X)^2} \Big[\partial_a var(Y|X,M,a)\partial_a f(a|X,M)+\frac{1}{2}var(Y|X,M,a) \partial^2_a f(a|X,M)\Big]\\
&+ \frac{\partial^2_a f(a'|X)}{2f(a'|X)^2}var[E(Y|X,M,a)|X,a']\\
&+  \Big(\gamma(X, M, a) - \eta(a, a', X)\Big)^2\times \Big[\frac{\partial^2_a f(M|X,a')}{2f(a'|X)f(M|X,A)} + \frac{\partial_a f(a'|X) \partial_a f(M|X,a')}{f(a'|X)^2 f(M|X,A)} \Big] \Bigg\}\\
E_3=&\mathbb{E}\bigg\{\frac{f(a'|M,X)}{f(a'|X)^2}\Big[\gamma(X,M,a)-\eta(a,a',X)\Big]\partial_a \gamma(X,M,a)\bigg\}\\
E_4 = & \mathbb{E}\bigg\{\frac{f(a'|M,X)}{f(a'|X)^2}\Big[\gamma(X,M,a)-\eta(a,a',X)\Big]\Big[\frac{1}{2}\partial^2_a \gamma(X,M,a) +\frac{\partial_a f(a|X,M)\partial_a \gamma(X,M,a)}{f(a|M,X)} \Big]\bigg\}\\
E_5=& \mathbb{E}\bigg\{\frac{f(a'|M,X)}{f(a'|X)^2}\Big[\gamma(X,M,a)-\eta(a,a',X)\Big]\\
&\qquad \times\Big[\frac{1}{6}\partial^3_a \gamma(X,M,a)+\frac{\partial_a f(a|X,M)\partial^2_a \gamma(X,M,a)}{2f(a|M,X)}+\frac{\partial^2_a f(a|X,M)\partial_a \gamma(X,M,a)}{2f(a|M,X)} \Big]\bigg\}
\end{align*}

Hence, as $h\rightarrow 0$, the asymptotic variance $V(a,a')$ converges to
$\int k(u)^2du\times E_1$.

Write $IF_k = m(O_k;\alpha,\lambda,\gamma,\psi_0(a,a'))$, raw moments as $\mu_l = \mathbb{E}(IF^l_k)$, and central moments as $\mu'_l = \mathbb{E}\Big\{(IF_k - \mathbb{E}(IF_k))^l\Big\}$. Note that $IF_k$ is also a function of $h$. For normality, we want to show that $IF_k$ satisfies the Lyapunov's condition for some $\delta>0$,

$$\bigg\{\sum^n_{k=1} var(IF_k) \bigg\}^{-(2+\delta)/2} \sum^n_{k=1} \mathbb{E}\Big[\Big|IF_k - \mathbb{E}(IF_k)\Big|^{2+\delta}\Big] \rightarrow 0$$

Previous derivation shows that $\mathbb{E}(IF_k) =\mu_1= O(h^2)$ and $var(IF_k)=\mu'_2 = O(\frac{1}{h^{d_A}})$.
From standard algebra, we can show that $\mu'_l = \mathbb{E}\Big\{(IF_k - \mathbb{E}(IF_k))^l\Big\} = O(\frac{1}{h^{(l-1)d_A}})$ and taking $\delta$ to be an even number,

$$\mathbb{E}\Big[\Big|IF_k - \mathbb{E}(IF_k)\Big|^{2+\delta}\Big] = \mathbb{E}\Big[\Big(IF_k - \mathbb{E}(IF_k)\Big)^{2+\delta}\Big]= O\Big(\frac{1}{h^{(1+\delta)d_A}}\Big).$$

\begin{align*}
&\bigg\{\sum^n_{k=1} var(IF_k) \bigg\}^{-(2+\delta)/2} \sum^n_{k=1} \mathbb{E}\Big[\Big|IF_k - \mathbb{E}(IF_k)\Big|^{2+\delta}\Big] \\
=&\bigg\{n O\Big(\frac{1}{h^{d_A}}\Big) \bigg\}^{-(2+\delta)/2} n O\Big(\frac{1}{h^{(1+\delta)d_A}}\Big)\\
=& O\Big(\frac{1}{\sqrt{(nh^{d_A})^\delta}}\Big) = o(1).
\end{align*}

Thus, the Lyapunov's condition is satisfied and by CLT,

$$\frac{\sum^n_{k=1}IF_k - n*\mathbb{E}(IF_k) }{\sqrt{n*var(IF_k)}} \sim N(0,1) \qquad \text{as }n\rightarrow \infty.$$

Therefore, $\sqrt{nh^{d_A}} (\hat{\psi}^{TR}(a,a')- \psi_0(a,a') -B(a,a')) \sim N(0, V(a,a'))$ as $n\rightarrow \infty$.

About the continuous DML paper:

“ The asymptotic variance is determined by $h\mathbb{E}\bigg\{\bigg[\frac{K_h(T-t)}{f_{T|X}(t|X)}(Y-\gamma(t,X)) \bigg]^2\bigg\}$. A standard algebra for kernel as above yields $V_t$.”

$$V_t = \mathbb{E}\bigg\{var[Y|T = t, X]/f_{T|X}(t|X)\bigg\}\int k(u)^2du$$

First. We know $B_t$ is the expectation of $\frac{K_h(T-t)}{f_{T|X}(t|X)}(Y-\gamma(t,X)) $, and $V_t$ is supposed to be the variance of the same term. Where does the $h$ in the quoted claim come from? I thought there should be
$$V_t = \mathbb{E}\bigg\{\bigg[\frac{K_h(T-t)}{f_{T|X}(t|X)}(Y-\gamma(t,X)) \bigg]^2\bigg\} - B^2_t.$$

Second. Suppose it is true that $V_t \approx h\mathbb{E}\bigg\{\bigg[\frac{K_h(T-t)}{f_{T|X}(t|X)}(Y-\gamma(t,X)) \bigg]^2\bigg\}$.

\begin{align*}
&h\mathbb{E}\bigg\{\bigg[\frac{K_h(T-t)}{f_{T|X}(t|X)}(Y-\gamma(t,X)) \bigg]^2\bigg\}\\
=& h\mathbb{E}\Bigg\{\mathbb{E}\bigg\{\bigg[\frac{K_h(T-t)}{f_{T|X}(t|X)}(Y-\gamma(t,X)) \bigg]^2\bigg|X\bigg\}\Bigg\}\\
=& h\mathbb{E}\Bigg\{\frac{1}{f_{T|X}(t|X)^2}\mathbb{E}\bigg\{K_h(T-t)^2(Y-\gamma(t,X) )^2\bigg|X\bigg\}\Bigg\}\\
=& h\mathbb{E}\Bigg\{\frac{1}{f_{T|X}(t|X)^2}\mathbb{E}\bigg\{K_h(T-t)^2\mathbb{E}\bigg[(Y-\gamma(t,X))^2 |T,X\bigg]\bigg|X\bigg\}\Bigg\}\\
=& h\mathbb{E}\Bigg\{\frac{1}{f_{T|X}(t|X)^2}\mathbb{E}\bigg\{K_h(T-t)^2\bigg[\mathbb{E}(Y^2|T,X)-2\gamma(T,X)\gamma(t,X)+\gamma(t,X)^2 \bigg]\bigg|X\bigg\}\Bigg\}\\
=& h\mathbb{E}\Bigg\{\frac{1}{f_{T|X}(t|X)^2}\mathbb{E}\bigg\{K_h(T-t)^2\bigg[var(Y|T,X)+\gamma(T,X)^2-2\gamma(T,X)\gamma(t,X)+\gamma(t,X)^2 \bigg]\bigg|X\bigg\}\Bigg\}\\
=& h\mathbb{E}\Bigg\{\frac{1}{f_{T|X}(t|X)^2}\mathbb{E}\bigg\{K_h(T-t)^2\bigg[var(Y|T,X)+(\gamma(T,X)-\gamma(t,X))^2 \bigg]\bigg|X\bigg\}\Bigg\}
\end{align*}

Because the $Y-\gamma(t,X)$ is squared, this does not directly lead to $\gamma(T,X)-\gamma(t,X)$ as in $B_t$ derivation.

The inner expectation

\begin{align*}
&\mathbb{E}\bigg\{K_h(T-t)^2\bigg[var(Y|T,X)+(\gamma(T,X)-\gamma(t,X))^2 \bigg]\bigg|X\bigg\}\\
=& \int K_h(T-t)^2\bigg[var(Y|T,X)+(\gamma(T,X)-\gamma(t,X))^2 \bigg]f_{T|X}(T|X) dT\\
=& \int \frac{1}{h^2} k^2(\frac{T-t}{h})\bigg[var(Y|T,X)+(\gamma(T,X)-\gamma(t,X))^2 \bigg]f_{T|X}(T|X) dT\\
=& \int \frac{1}{h^2} k^2(u)\bigg[var(Y|t+uh,X)+(\gamma(t+uh,X)-\gamma(t,X))^2 \bigg]f_{T|X}(t+uh|X) hdu\\
=& \frac{1}{h}\int  k^2(u)\bigg[var(Y|t+uh,X)+(\gamma(t+uh,X)-\gamma(t,X))^2 \bigg]f_{T|X}(t+uh|X) du\\
=& \frac{1}{h}\int  k^2(u)\bigg[var(Y|t+uh,X)+\bigg(uh\partial_t\gamma(t,X) + \frac{u^2h^2}{2} \partial^2_t \gamma(t,X)+ O(h^3)\bigg)^2  \bigg]\\
&\times \bigg(f_{T|X}(t|X)+uh\partial_t f_{T|X}(t|X)+u^2 h^2\partial^2_t f_{T|X}(t|X)+O(h^3)\bigg) du\\
=& \frac{1}{h}\int  k^2(u)\bigg[\bigg(var(Y|t,X)+ uh\partial_t var(Y|t,X)+\frac{u^2 h^2}{2}\partial^2_t var(Y|t,X)+O(h^3)\bigg)\\
&+\bigg(uh\partial_t\gamma(t,X) + \frac{u^2h^2}{2} \partial^2_t \gamma(t,X)+ O(h^3)\bigg)^2  \bigg]\\
&\times \bigg(f_{T|X}(t|X)+uh\partial_t f_{T|X}(t|X)+\frac{u^2 h^2}{2}\partial^2_t f_{T|X}(t|X)+O(h^3)\bigg) du\\
\end{align*}

By $\int k^2(u)u du =0$, the above term equals

\begin{align*}
& \frac{1}{h}var(Y|t,X)f_{T|X}(t|X)\int k^2(u)du+O(h)\\
\end{align*}

Then, the asymptotic variance

\begin{align*}
&h^{d_A}\mathbb{E}\bigg\{\bigg[\frac{K_h(T-t)}{f_{T|X}(t|X)}(Y-\gamma(t,X)) \bigg]^2\bigg\}\\
=& h^{d_A}\mathbb{E}\Bigg\{\frac{1}{f_{T|X}(t|X)^2}\frac{1}{h}var(Y|t,X)f_{T|X}(t|X)\int k^2(u)du+O(h)\Bigg\}\\
=&h^{d_A-1}\mathbb{E}\bigg\{var[Y|T = t, X]/f_{T|X}(t|X)\bigg\}\int k(u)^2du + O(h^{d_A+1})
\end{align*}