# Asymptotic expected sensitivity function and its applications to
nonparametric correlation estimators

Qingyang Zhang
Department of Mathematical Sciences, University of Arkansas, AR 72701
Email<EMAIL_ADDRESS>
###### Abstract

We introduce a new type of influence function, the asymptotic expected
sensitivity function, which is often equivalent to but mathematically more
tractable than the traditional one based on the Gâteaux derivative. To
illustrate, we study the robustness of some important rank correlations,
including Spearman’s and Kendall’s correlations, and the recently developed
Chatterjee’s correlation.

Keywords: Robustness measure; influence function; sensitivity function;
Chatterjee’s correlation

##  1 Introduction

In the literature of robust statistics, various measures of robustness have
been proposed, including influence function, sensitivity function and
breakdown point [1, 2, 3]. These measures provide a means to assess the
robustness of statistical functionals and estimators, particularly in the face
of outliers or deviations from the underlying model. Recently, these
robustness measures have found applications in machine learning as well [4, 5,
6]. For instance, by quantifying the impact of individual training data points
on a model’s predictions, these robustness measures may provide insights into
how specific samples influence the model’s decision-making process. This
capability enables the identification of critical data points that
significantly affect the model’s performance, facilitating more informed data
curation and model debugging.

In this work, we focus on two key measures: the influence function for
functionals and sensitivity function for estimators. We begin with the
definitions and a brief discussion on their respective limitations. For random
variable $X$ with cumulative distribution function (c.d.f.) $F$, the influence
function ($IF$) of a statistical functional $R(F)$ at $X=x$ is defined as

$\mbox{IF}(x,~{}R,~{}F)=\lim_{\epsilon\downarrow
0}\frac{R[(1-\epsilon)F+\epsilon\Delta_{x}]-R(F)}{\epsilon},$

where $\Delta_{x}$ is a Dirac distribution which puts all its mass on $X=x$.
The influence function $\mbox{IF}(x,~{}R,~{}F)$ is essentially a Gâteaux
derivative at $X=x$, which quantifies the sensitivity of $R(F)$ to
infinitesimal modifications in a single data point $x$. A substantial
influence function indicates that the functional is highly susceptible,
whereas a modest influence function suggests that the functional is more
robust to $X=x$. For all linear functionals, i.e., functionals that can be
written as $R(F)=\int g(t)dF(t)$, the influence function is $g(x)-R(F)$.

Sensitivity function (SF), sometimes also called empirical influence function,
can be viewed as a sample-based counterpart of influence function. Let $F_{n}$
be the empirical distribution function, and $R_{n}$ be an estimator of $R$
based on i.i.d. samples $X_{1},~{}...,~{}X_{n}$. An estimator $R_{n}$ is
called a plug-in estimator if $R_{n}=R(F_{n})$. The SF of $R_{n}$ at $X=x$ is
defined as

$\mbox{SF}(x,~{}R_{n},~{}F_{n})=(n+1)\left[R_{n+1}(X_{1},~{}...,~{}X_{n},~{}x)-R_{n}(X_{1},~{}...,~{}X_{n})\right],$

which represents the change in estimator when a new observation $x$ is added
to the existing sample of size $n$. The asymptotic version of sensitivity
function (ASF) is defined as
$\lim_{n\rightarrow\infty}\mbox{SF}(x,~{}R_{n},~{}F_{n})$. It is noteworthy
that the above definitions of IF, SF and ASF are extendable to any dimension.

Though the influence function and sensitivity function have been widely used
as measures of robustness, they both have certain limitations from practical
point of view. For instance, the influence function is based on Gâteaux
derivative, which can be difficult to derive for some complicated functionals.
Consider Dette-Siburg-Stoimenov’s dependence measure [7], also known as the
limit of Chatterjee’s rank correlation [8], expressed as

$\xi(X,Y)=\frac{\int V(\mathbb{E}(\mathbbm{1}\\{Y\geq t|X\\}))dF_{Y}(t)}{\int
V(\mathbbm{1}\\{Y\geq t\\})dF_{Y}(t)}.$

This functional, characterized by a ratio of integrals, presents a relatively
intricate structure. The robustness of $\xi(X,Y)$ remains unexplored due to
the difficulty of deriving its Gâteaux derivative.

The sensitivity function is generally more straightforward to derive, but it
relies on the random samples, making it not suitable as a population-level
measure. For certain consistent estimators, the asymptotic sensitivity
function (obtained by letting sample size approach infinity) are deterministic
and equivalent to the influence function. However, this equivalence does not
hold for many other functionals and estimators. Notably, Croux (1998)
demonstrated that the asymptotic sensitivity function of the sample median,
rather than being deterministic, follows an exponential-type distribution [9].

Motivated by both Croux’s work and Dette-Siburg-Stoimenov’s dependence
measure, we propose a new robustness measure based on the asymptotic
expectation of sensitivity function, termed asymptotic expected sensitivity
function (AESF). For many functionals, the proposed measure can be derived
similarly to sensitivity function, without requiring the Gâteaux derivative.
Crucially, unlike the asymptotic sensitivity function, the AESF is always
deterministic, and we demonstrate that in many cases, it coincides with the
influence function.

The remainder of the paper is structured as follows: Section 2 introduces the
new robustness measure with some elementary examples, and then discusses its
connections to IF, SF. In Section 3, we apply the proposed measure to analyze
the robustness of three important rank correlations, including Spearman’s,
Kendall’s and Chatterjee’s correlations. Section 4 discusses and concludes the
paper.

##  2 A new influence function

In this section, we define the new influence function AESF, followed by some
illustrative examples. We then briefly discuss the relationships between AESF,
ASF and IF, shedding light on their interconnectedness. For clarity, we focus
on univariate case, but it is important to note that the new measure can be
readily extended to multivariate settings.

###  2.1 Definition and elementary examples

Consider the expected sensitivity function, expressed as

$\mbox{ESF}(x,~{}R,~{}F,~{}n)=(n+1)\mathbb{E}\left[R_{n+1}(X_{1},~{}...,~{}X_{n},~{}x)-R_{n}(X_{1},~{}...,~{}X_{n})\right].$

The expectation in ESF is to eliminate the randomness of $SF$. In spirit, it
is similar to the $SF$ based on stylized sample proposed by Andrews et al.
(1972) (page 96, [10]), which is essentially a pseudo-sample consisting of the
$n$ expected normal order statistics from $(X_{1},~{}...,~{}X_{n})$. However,
the $ESF$ distinguishes itself with more tractable theoretical properties.

The asymptotic expected sensitivity function (AESF) can be then defined as

$\mbox{AESF}(x,~{}R,~{}F)=\lim_{n\rightarrow\infty}\mbox{ESF}(x,~{}R,~{}F,~{}n).$

It is not hard to see that AESF is always deterministic. While $AESF$ often
coincides with the expectation of ASF for numerous functionals and their
estimators, their general equivalence is not guaranteed, unless for uniformly
bounded sensitivity function where the Lebesgue’s Dominated Convergence
Theorem applies. We use the following elementary examples to elucidate the
derivation of AESF and its relationship to ASF and IF.

#### Example 1: Mean

The expectation functional, $\mu(F)=\int_{-\infty}^{\infty}tdF(t)$, is a
linear functional and its influence function is
$\mbox{IF}(x,~{}\mu,~{}F)=x-\mu$, which is unbounded. The sample mean can be
treated as a plug-in estimator of $\mu(F)$, i.e.,
$\mu_{n}=\mu(F_{n})=\sum_{i=1}^{n}X_{i}/n$. The sensitivity function of
$\mu_{n}$ is

$\displaystyle\mbox{SF}(x,~{}\mu_{n},~{}F_{n})$
$\displaystyle=(n+1)\left[\frac{x+\sum_{i=1}^{n}X_{i}}{n+1}-\frac{\sum_{i=1}^{n}X_{i}}{n}\right]$
$\displaystyle=x-\frac{\sum_{i=1}^{n}X_{i}}{n}.$

By Strong Law of Large Numbers (SLLN), $\sum_{i=1}^{n}X_{i}/n$ converges to
$\mu$ almost surely, therefore for any given $x$,
$\mbox{SF}(x,~{}\mu_{n},~{}F_{n})$ converges to $x-\mu$ almost surely.

The expected sensitivity function is

$\displaystyle\mbox{ESF}(x,~{}\mu,~{}F,~{}n)$
$\displaystyle=\mathbb{E}[\mbox{SF}(x,~{}R_{n},~{}F_{n})]$
$\displaystyle=x-\frac{\sum_{i=1}^{n}E(X_{i})}{n}$ $\displaystyle=x-\mu,$

which does not depend on $n$, and $\mbox{AESF}(x,~{}\mu,~{}F)=x-\mu$.
Therefore, we have $\mbox{AESF}(x,~{}\mu,~{}F)=\mbox{IF}(x,~{}\mu,~{}F)$, and
$\mbox{SF}(x,~{}\mu_{n},~{}F_{n})$ converges almost surely to
$\mbox{AESF}(x,~{}\mu,~{}F)$ for any $x$.

#### Example 2: Variance

The variance functional,
$\sigma^{2}(F)=\int_{-\infty}^{\infty}[t-\int_{-\infty}^{\infty}sdF(s)]^{2}dF(t)$,
has the influence function
$\mbox{IF}(x,~{}\sigma^{2},~{}F)=(x-\mu)^{2}-\sigma^{2}$, which is unbounded.

The sensitivity function of sample variance $\sigma^{2}_{n}$ is

$\displaystyle\mbox{SF}(x,~{}\sigma^{2}_{n},~{}F_{n})$
$\displaystyle=(n+1)\left[\frac{x^{2}+\sum_{i=1}^{n}X^{2}_{i}}{n+1}-\left(\frac{x+\sum_{i=1}^{n}X_{i}}{n+1}\right)^{2}-\frac{\sum_{i=1}^{n}X^{2}_{i}}{n}+\left(\frac{\sum_{i=1}^{n}X_{i}}{n}\right)^{2}\right]$
$\displaystyle=\frac{n}{n+1}x^{2}-\frac{\sum_{i=1}^{n}X_{i}}{n}\frac{2n}{n+1}x+\frac{2n+1}{n+1}\left(\frac{\sum_{i=1}^{n}X_{i}}{n}\right)^{2}-\frac{\sum_{i=1}^{n}X^{2}_{i}}{n}.$

By SLLN, for any $x$, $\mbox{SF}(x,\sigma^{2}_{n},F_{n})$ converges to
$(x-\mu)^{2}-\sigma^{2}$ almost surely.

The expected sensitivity function is

$\displaystyle\mbox{ESF}(x,~{}\sigma^{2},~{}F,~{}n)$
$\displaystyle=\mathbb{E}\left[\mbox{SF}(x,~{}\sigma^{2}_{n},~{}F_{n})\right]$
$\displaystyle=\frac{n}{n+1}x^{2}-\frac{2n}{n+1}x\mu+\frac{n}{n+1}\mu^{2}-\frac{n^{2}-n-1}{n(n+1)}\sigma^{2},$

and $\mbox{AESF}(x,~{}\sigma^{2},~{}F)=(x-\mu)^{2}-\sigma^{2}$. Therefore we
have $\mbox{AESF}(x,~{}\sigma^{2},~{}F)=\mbox{IF}(x,~{}\sigma^{2},~{}F)$, and
$\mbox{SF}(x,~{}\sigma^{2}_{n},~{}F_{n})$ converges almost surely to
$\mbox{AESF}(x,~{}\sigma^{2},~{}F)$ for any $x$.

#### Example 3: Maximum of uniform distribution

Let $F$ be a uniform distribution on $[0,\theta]$, we consider a simple
functional $R(F)=\inf\\{t|F(t)=1\\}=\theta$, and the plug-in estimator
$R_{n}=R(F_{n})=\inf\\{t|F_{n}(t)=1\\}=X_{(n)}$, where $X_{(1)}\leq...\leq
X_{(n)}$. For any $0\leq\epsilon<1$ and $0\leq x\leq\theta$, we have
$R[(1-\epsilon)F+\epsilon\Delta_{x}]=\theta$, thus

$\mbox{IF}(x,~{}R,~{}F)=0\text{~{}~{}for~{}~{}}0\leq x\leq\theta.$

The sensitivity function of $R_{n}$ is

$\mbox{SF}(x,~{}R_{n},~{}F_{n})=(n+1)(x-X_{(n)})\mathbf{1}\left[X_{(n)}<x\right].$

As $X_{(n)}\xrightarrow[]{a.s.}\theta$, for any $x<\theta$,
$\mbox{SF}(x,~{}R_{n},~{}F_{n})\xrightarrow[]{a.s.}0$. For $x=\theta$, the
c.d.f. of $\mbox{SF}(x,~{}R_{n},~{}F_{n})$ is

$\displaystyle\mbox{P}\left[\mbox{SF}(x,~{}R_{n},~{}F_{n})<s\right]$
$\displaystyle=1-\mbox{P}\left[X_{(n)}<\theta-\frac{s}{n+1}\right]$
$\displaystyle=1-\left[1-\frac{s}{\theta(n+1)}\right]^{n},$

which converges in distribution to an exponential distribution with rate
parameter $\theta$, i.e., the ASF is not deterministic.

Furthermore, because the c.d.f. of $X_{(n)}$ is
$F_{X_{(n)}}(x)=(x/\theta)^{n}$ for $0\leq x\leq\theta$, we have

$\displaystyle\mbox{ESF}(x,~{}R,~{}F,~{}n)$
$\displaystyle=\mathbb{E}\left[(n+1)(x-X_{(n)})\mathbf{1}\left(X_{(n)}<x\right)\right]$
$\displaystyle=\mathbb{E}\left[x(n+1)\mathbf{1}\left(X_{(n)}<x\right)\right]-\mathbb{E}\left[(n+1)X_{(n)}\mathbf{1}\left(X_{(n)}<x\right)\right]$
$\displaystyle=x(n+1)\left(\frac{x}{\theta}\right)^{n}-xn\left(\frac{x}{\theta}\right)^{n}$
$\displaystyle=x\left(\frac{x}{\theta}\right)^{n},$

and

$\mbox{AESF}(x,~{}R,~{}F)=\begin{cases}0,&0\leq x<\theta\\\
\theta,&x=\theta.\end{cases}$

From the derivations above, it can be seen that the $IF$, $AESF$ and $ASF$
coincide for $0\leq x<\theta$, but they diverge notably at $x=\theta$.
Specifically, we observe the following

  * •

$\mbox{IF}(x=\theta,~{}R,~{}F)=0$

  * •

$\mbox{AESF}(x=\theta,~{}R,~{}F)=\theta$

  * •

$\lim_{n\rightarrow\infty}\mbox{SF}(x=\theta,~{}R_{n},~{}F_{n})$ is not
deterministic.

This example highlights the potential for significant discrepancies between
these metrics for certain nonlinear functionals and specific values of $x$.
The subsequent section will explore the relationships between these
interconnected metrics and establish sufficient conditions under which they
achieve equivalence.

###  2.2 Relationships between AESF, SF and IF

For any linear functional $R(F)=\int g(t)dF(t)$ with
$|\mathbb{E}[g(X)]|<\infty$, let $R_{n}$ be the plug-in estimator, i.e.,
$R_{n}=\int g(t)dF_{n}(t)=\sum_{i=1}^{n}g(X_{i})/n$, where
$X_{1},~{}...,~{}X_{n}$ are $i.i.d.$ samples, we have

$\displaystyle\mbox{IF}(x,~{}R,~{}F)$ $\displaystyle=g(x)-\int g(t)dF(t),$
$\displaystyle\mbox{SF}(x,~{}R_{n},~{}F_{n})$
$\displaystyle=g(x)-\frac{1}{n}\sum_{i=1}^{n}g(X_{i}).$

As $\sum_{i=1}^{n}g(X_{i})/n\xrightarrow[]{a.s.}\int g(t)dF(t)$, we have
$\mbox{SF}(x,~{}R_{n},~{}F_{n})\xrightarrow[]{a.s.}g(x)-\int
g(t)dF(t)=\mbox{IF}(x,~{}R,~{}F)$. In addition,
$\mbox{ESF}(x,~{}R,~{}F,~{}n)=g(x)-\sum_{i=1}^{n}\mathbb{E}\left[g(X_{i})\right]/n=g(x)-\int
g(t)dF(t)=\mbox{IF}(x,~{}R,~{}F)$, we have for any $x$

$\mbox{SF}(x,~{}R_{n},~{}F_{n})\xrightarrow[]{a.s.}\mbox{AESF}(x,~{}R,~{}F)=\mbox{IF}(x,~{}R,~{}F).$

Because the expectation functional $\mu(F)$ is linear, we have
$\mbox{SF}(x,~{}\mu_{n},~{}F_{n})\xrightarrow[]{a.s.}\mbox{AESF}(x,~{}\mu,~{}F)=\mbox{IF}(x,~{}\mu,~{}F)$
for any $x$. The following theorem extends the results for linear functionals
to any continuous and bounded transformation (a detailed proof is provided in
the Appendix).

######  Theorem 1.

Consider a statistical functional $R(F)=\phi\left[\int g(t)dF(t)\right]$ and
its plug-in estimator $R_{n}=\phi\left[\int g(t)dF_{n}(t)\right]$. Suppose
$\int|g(t)|dF(t)<\infty$, $\phi$ is differentiable with continuous and bounded
$\phi^{{}^{\prime}}$, then for any $x$, we have

$\mbox{SF}(x,~{}R_{n},~{}F_{n})\xrightarrow[]{a.s.}\mbox{AESF}(x,~{}R,~{}F)=\mbox{IF}(x,~{}R,~{}F)=\left[g(x)-\int
g(t)dF(t)\right]\phi^{{}^{\prime}}\left[\int g(t)dF(t)\right].$

The following corollary can be easily obtained by a linear combination:

######  Corollary 1.

If $[\phi_{l}(\cdot),~{}g_{l}(\cdot)]_{l=1,~{}...,~{}L}$ all satisfy the
conditions in Theorem 1, $(a_{1},~{}...,~{}a_{L})$ are constants, then for
functional $R(F)=\sum_{l=1}^{L}a_{l}\phi_{l}\left[\int g_{l}(t)dF(t)\right]$,
we have

$\mbox{SF}(x,~{}R_{n},~{}F_{n})\xrightarrow[]{a.s.}\mbox{AESF}(x,~{}R,~{}F)=\mbox{IF}(x,~{}R,~{}F)=\sum_{l=1}^{L}a_{l}\left[g_{l}(x)-\int
g_{l}(t)dF(t)\right]\phi_{l}^{{}^{\prime}}\left[\int g_{l}(t)dF(t)\right].$

By Corollary 1, it is immediate that for variance, we have
$\mbox{SF}(x,~{}\sigma^{2}_{n},~{}F_{n})\xrightarrow[]{a.s.}\mbox{AESF}(x,~{}\sigma^{2},~{}F)=\mbox{IF}(x,~{}\sigma^{2},~{}F)$
for any $x$, simply by letting
$a_{1}=1,~{}a_{2}=-1,~{}g_{1}(t)=t^{2},~{}g_{2}(t)=t,~{}\phi_{1}(z)=z,~{}\phi_{2}(z)=z^{2}$.

For functionals that cannot be expressed as continuous transformations of
linear functionals, equivalence results are not generally guaranteed. An open
question for this is the identification of sufficient and necessary conditions
for such equivalence.

##  3 Application to some rank correlations

In this section, we derive the $AESF$ of some important nonparametric
correlations, including the widely used Kendall’s and Spearman’s correlations
and a recent one developed by Chatterjee (2021). It is found that for
Kendall’s and Spearman’s correlations, the $AESF$ is same as the $IF$. For
Chatterjee’s correlation, the $IF$ is unknown due to the complicated form of
its functional representation, a.k.a., Dette-Siburg-Stoimenov’s dependence
measure, but the derivation of $AESF$ is tractable, and we use $AESF$ as an
alternative metric to study its robustness under various correlative patterns.

###  3.1 Kendall’s and Spearman’s correlations

Let $(X_{1},~{}Y_{1}),~{}...,~{}(X_{n},~{}Y_{n})$ be $n$ i.i.d. samples of
$(X,~{}Y)$. The Kendall’s correlation is defined as

$\tau_{n}=\frac{2}{n(n-1)}\sum_{i<j}\mbox{sgn}\left[(X_{i}-X_{j})(Y_{i}-Y_{j})\right],$

where $\mbox{sgn}(t)$ equals 1 for $t\geq 0$ and -1 for $t<0$. The functional
representation of $\tau_{n}$ is

$\tau=\mathbb{E}\left\\{\mbox{sgn}[(X-X^{{}^{\prime}})(Y-Y^{{}^{\prime}})]\right\\},$

where $(X^{{}^{\prime}},Y^{{}^{\prime}})$ represents an independent copy of
$(X,Y)$. It can be seen that Kendall’s correlation is a U-statistic, and its
sensitivity function is

$\displaystyle\mbox{SF}[(x,~{}y),~{}\tau_{n},~{}F_{n}]$
$\displaystyle=(n+1)\left\\{\frac{2}{n(n+1)}\sum_{i<j}\mbox{sgn}\left[(X_{i}-X_{j})(Y_{i}-Y_{j})\right]+\frac{2}{n(n+1)}\sum_{i=1}^{n}\mbox{sgn}\left[(X_{i}-x)(Y_{i}-y)\right]-\tau_{n}\right\\}$
$\displaystyle=\frac{2}{n}\sum_{i=1}^{n}\mbox{sgn}\left[(X_{i}-x)(Y_{i}-y)\right]-\frac{4}{n(n-1)}\sum_{i<j}\mbox{sgn}\left[(X_{i}-X_{j})(Y_{i}-Y_{j})\right].$

Note that $\mbox{sgn}(t)=2\cdot\mathbf{1}(t\geq 0)-1$, we have

$\displaystyle\mathbb{E}\left\\{\mbox{SF}[(x,~{}y),~{}\tau_{n},~{}F_{n}]\right\\}$
$\displaystyle=4P[(X-x)(Y-y)>0]-2-\mathbb{E}\left\\{\mbox{sgn}[(X-X^{{}^{\prime}})(Y-Y^{{}^{\prime}})]\right\\}$
$\displaystyle=4P[(X-x)(Y-y)>0]-2-\tau.$

The $ESF$ does not rely on sample size $n$, therefore

$\mbox{AESF}[(x,~{}y),~{}\tau,~{}F]=4P[(X-x)(Y-y)>0]-2-\tau,$

which is equivalent to the $\mbox{IF}[(x,~{}y),~{}\tau,~{}F]$ (Proposition 1,
[11]), and uniformly bounded by 3. As a special case, for bivariate normal
distribution with correlation $\rho$, we have

$\mbox{AESF}[(x,~{}y),~{}\tau,~{}F]=8\Phi_{\rho}(x,y)-4\Phi(x)-4\Phi(y)+2-\frac{2}{\pi}\arcsin(\rho).$

Figure 1 below demonstrates the $AESF$ values for a standard bivariate normal
distribution with $\rho=0.7$.

[Figure 1 about here]

Next we derive the $AESF$ for Spearman’s correlation. The correlation
estimator is

$S_{n}=1-\frac{6\sum_{i=1}^{n}[r(X_{i})-r(Y_{i})]^{2}}{n(n-1)(n+1)},$

where $r(X_{i})=\sum_{j=1}^{n}\mathbb{I}(X_{j}\leq X_{i})$ and
$r(Y_{i})=\sum_{j=1}^{n}\mathbb{I}(Y_{j}\leq Y_{i})$. The functional
representation of $S_{n}$ is

$S=12\mathbb{E}[F_{X}(X)F_{Y}(Y)]-3.$

Let $r^{*}(X_{i})$ and $r^{*}(Y_{i})$ be the ranks of $X_{i}$ and $Y_{i}$
after $(x,y)$ is added, the $SF$ can be decomposed as follows

$\mbox{SF}[(x,~{}y),~{}S_{n},~{}F_{n}]=\Delta_{n1}+\Delta_{n2}+\Delta_{n3},$

where

$\displaystyle\Delta_{n1}$
$\displaystyle=\frac{6\sum_{i=1}^{n}[r(X_{i})-r(Y_{i})]^{2}-6\sum_{i=1}^{n}[r^{*}(X_{i})-r^{*}(Y_{i})]^{2}}{n(n+2)},$
$\displaystyle\Delta_{n2}$
$\displaystyle=-\frac{6[r^{*}(x)-r^{*}(y)]^{2}}{n(n+2)},$
$\displaystyle\Delta_{n3}$
$\displaystyle=\frac{18\sum_{i=1}^{n}[r(X_{i})-r(Y_{i})]^{2}}{n(n-1)(n+2)}.$

First, for $\Delta_{n2}$, we have

$\Delta_{n2}=\frac{12r^{*}(x)r^{*}(y)}{(n+1)^{2}}+O(1/n).$

By Glivenko-Cantelli Theorem

$\lim_{n\rightarrow\infty}\Delta_{n2}=12F_{X}(x)F_{Y}(y),~{}a.s.$

In addition, as $[r^{*}(x)-r^{*}(y)]^{2}\leq(n-1)^{2}$, $\Delta_{n2}$ is
uniformly bounded by 6. By Lebesgue’s Dominated Convergence Theorem

$\lim_{n\rightarrow\infty}\mathbb{E}(\Delta_{n2})=\mathbb{E}(\lim_{n\rightarrow\infty}\Delta_{n2})=12F_{X}(x)F_{Y}(y).$

For $\Delta_{n3}$, it is easy to see that

$\Delta_{n3}=\frac{3(n+1)}{n+2}(1-S_{n}),~{}\text{and}~{}0<\Delta_{n3}<6.$

By Lebesgue’s Dominated Convergence Theorem

$\lim_{n\rightarrow\infty}\mathbb{E}(\Delta_{n3})=\mathbb{E}(\lim_{n\rightarrow\infty}\Delta_{n3})=3(1-S).$

Last for $\Delta_{n1}$, we have

$\displaystyle\Delta_{n1}$
$\displaystyle=\frac{12\sum_{i=1}^{n}[r^{*}(X_{i})r^{*}(Y_{i})-r(X_{i})r(Y_{i})]}{n(n+2)}-12+O(1/n)$
$\displaystyle=\frac{12}{n(n+2)}\sum_{i=1}^{n}r(X_{i})\mathbf{1}(Y_{i}\geq
y)+\frac{12}{n(n+2)}\sum_{i=1}^{n}r(Y_{i})\mathbf{1}(X_{i}\geq x)-12+O(1/n).$

Therefore

$\displaystyle\mathbb{E}(\Delta_{n1})$
$\displaystyle=12\mathbb{E}[F_{X}(X)\mathbf{1}(Y\geq
y)]+12\mathbb{E}[F_{Y}(Y)\mathbf{1}(X\geq x)]-12+O(1/n),$
$\displaystyle\lim_{n\rightarrow\infty}\mathbb{E}(\Delta_{n1})$
$\displaystyle=12\mathbb{E}[F_{X}(X)\mathbf{1}(Y\geq
y)]+12\mathbb{E}[F_{Y}(Y)\mathbf{1}(X\geq x)]-12.$

Summarizing the results above, we have

$\mbox{AESF}[(x,~{}y),~{}S,~{}F]=12F_{X}(x)F_{Y}(y)+12\mathbb{E}[F_{X}(X)\mathbf{1}(Y\geq
y)]+12\mathbb{E}[F_{Y}(Y)\mathbf{1}(X\geq x)]-3S-9,$

which is equivalent to the $\mbox{IF}[(x,y),~{}S,~{}F]$ (Proposition 1, [11]),
and uniformly bounded. As a special case, for bivariate normal distribution
with correlation $|\rho|<1$, we have

$\displaystyle\mbox{AESF}[(x,~{}y),~{}S,~{}F]=$ $\displaystyle
12\Phi(x)\Phi(y)+12\left\\{\mathbb{E}\left[\Phi(X)\Phi\left(\frac{\rho
X-y}{\sqrt{1-\rho^{2}}}\right)\right]+\mathbb{E}\left[\Phi(Y)\Phi\left(\frac{\rho
Y-x}{\sqrt{1-\rho^{2}}}\right)\right]\right\\}$
$\displaystyle-\frac{18}{\pi}\arcsin\left(\frac{\rho}{2}\right)-9.$

Figure 2 below demonstrates the $AESF$ for a standard bivariate normal
distribution with $\rho=0.7$. Figure 3 presents the comparison of Kendall’s
and Spearman’s correlation under the same setting, where it can be seen that
Kendall’s correlation is more robust (smaller absolute $AESF$) than Spearman’s
correlation for most of the discordant pairs (red), but less robust to the
concordant pairs.

[Figure 2 about here]

[Figure 3 about here]

###  3.2 Chatterjee correlation

Chatterjee (2021) introduced a new correlation test, which is distribution-
free, asymptotically normal, and consistent against all alternatives [8]. Due
to its nice properties, Chatterjee’s correlation has attracted much attention
over the past two years [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24].
However, the robustness of this new metric remains unexplored, partially
because the influence function of its functional representation is difficult
to derive.

Here, we study the robustness of Chatterjee’s correlation using the proposed
$AESF$. We begin with the definition of this new correlation measure. Let $X$
and $Y$ be two continuous variables, and $(X_{i},~{}Y_{i})_{i=1,...,n}$ be n
$i.i.d.$ samples. Assuming that $X_{i}$’s and $Y_{i}$’s have no ties, the data
can be uniquely arranged as $(X_{(1)},Y_{(1)}),~{}...,~{}(X_{(n)},Y_{(n)})$,
such that $X_{(1)}<\cdots<X_{(n)}$. Here $Y_{(1)},~{}...,~{}Y_{(n)}$ denote
the concomitants. Let $r_{i}$ be the rank of $Y_{(i)}$, i.e.,
$r_{i}=\sum_{k=1}^{n}\mathbbm{1}\\{Y_{(k)}\leq Y_{(i)}\\}$, Chatterjee’s
correlation $\xi_{n}(X,Y)$ is defined as

$\xi_{n}(X,Y)=1-\frac{3\sum_{i=1}^{n-1}|r_{i+1}-r_{i}|}{n^{2}-1}.$ (1)

The asymptotic behavior of $\xi_{n}(X,Y)$ and related problems have been
examined in recent papers. For instance, in his original paper [8], Chatterjee
showed that

$\xi_{n}(X,Y)\rightarrow\xi(X,Y)=\frac{\int V(E(\mathbbm{1}\\{Y\geq
t|X\\}))dF_{Y}(t)}{\int V(\mathbbm{1}\\{Y\geq t\\})dF_{Y}(t)},~{}a.s.$

The limiting quantity $\xi(X,Y)$ is also known as Dette-Siburg-Stoimenov’s
dependence measure [7], which is between 0 and 1 (0 if and only if $X$ and $Y$
are independent, 1 if and only if $Y$ is a measurable function of $X$).

Let $(r^{*}_{1},...,r^{*}_{n+1})$ be the ranks of concomitants after $(x,y)$
being added, we have

$\displaystyle\mbox{SF}[(x,~{}y),~{}\xi_{n},~{}F_{n}]$
$\displaystyle=(n+1)\bigg{\\{}\xi_{n+1}[(X_{1},~{}Y_{1}),...,~{}(X_{n},~{}Y_{n}),~{}(x,~{}y)]-\xi_{n}[(X_{1},~{}Y_{1}),~{}...,~{}(X_{n},~{}Y_{n})]\bigg{\\}}$
$\displaystyle=3(n+1)\left\\{\frac{\sum_{i=1}^{n-1}|r_{i+1}-r_{i}|}{n^{2}-1}-\frac{\sum_{j=1}^{n}|r^{*}_{j+1}-r^{*}_{j}|}{(n+1)^{2}-1}\right\\}$
$\displaystyle=\frac{3\sum_{i=1}^{n-1}|r_{i+1}-r_{i}|}{n-1}-\left(1-\frac{2n+1}{n^{2}+2n}\right)\frac{\sum_{j=1}^{n}|r^{*}_{j+1}-r^{*}_{j}|}{n-1}.$

Further, let

$\Delta_{n}=\frac{3\sum_{j=1}^{n}|r^{*}_{j+1}-r^{*}_{j}|}{n-1}-\frac{3\sum_{i=1}^{n-1}|r_{i+1}-r_{i}|}{n-1},$

the sensitivity function can be written as

$\mbox{SF}[(x,~{}y),~{}\xi_{n},~{}F_{n}]=\frac{2n+1}{n^{2}+2n}\frac{3\sum_{i=1}^{n-1}|r_{i+1}-r_{i}|}{n-1}-\Delta_{n}\left(1-\frac{2n+1}{n^{2}+2n}\right).$
(2)

Let $N(i)$ be the index $j$ such that $X_{j}$ is immediately to the right of
$X_{i}$ if we arrange the $X$’s in increasing order. If $X_{i}$ is the
rightmost value, define $N(i)$ arbitrarily, which does not matter since the
contribution of a single term in the above sum is $O(1/n)$. For the first term
in Equation (2), we have

$\lim_{n\rightarrow\infty}\mathbb{E}\left[\frac{\sum_{i=1}^{n-1}|r_{i+1}-r_{i}|}{n^{2}}\right]=\lim_{n\rightarrow\infty}\mathbb{E}\left[\frac{1}{n}\sum_{i=1}^{n}|F_{Yn}(Y_{i})-F_{Yn}(Y_{N(i)})|\right].$

Let

$\displaystyle Q_{n}$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}|F_{Yn}(Y_{i})-F_{Yn}(Y_{N(i)})|,$
$\displaystyle Q^{{}^{\prime}}_{n}$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}|F_{Y}(Y_{i})-F_{Y}(Y_{N(i)})|,$

and

$\delta_{n}=\sup_{a\in\mathbb{R}}|F_{Yn}(a)-F_{Y}(a)|,$

we have

$|Q^{{}^{\prime}}_{n}-Q_{n}|<2\delta_{n}.$

By Glivenko-Cantelli Theorem, $\delta_{n}\rightarrow 0$ almost surely as
$n\rightarrow\infty$. Since $0\leq\delta_{n}\leq 1$, we have

$\lim_{n\rightarrow\infty}\mathbb{E}|Q^{{}^{\prime}}_{n}-Q_{n}|=0,$

and

$\displaystyle\lim_{n\rightarrow\infty}\mathbb{E}\left[\frac{\sum_{i=1}^{n-1}|r_{i+1}-r_{i}|}{n^{2}}\right]$
$\displaystyle=\lim_{n\rightarrow\infty}\mathbb{E}\left[\frac{1}{n}\sum_{i=1}^{n}|F_{Y}(Y_{i})-F_{Y}(Y_{N(i)})|\right]$
$\displaystyle=1-2\lim_{n\rightarrow\infty}\mathbb{E}\left[\frac{1}{n}\sum_{i=1}^{n}\min(F(Y_{i}),F(Y_{N(i)}))\right].$

By Lemma (9.10) of [8], we have

$\lim_{n\rightarrow\infty}\mathbb{E}\left[\frac{\sum_{i=1}^{n-1}|r_{i+1}-r_{i}|}{n^{2}}\right]=1-2\mathbb{E}_{Y^{\prime}}\mathbb{E}_{X}\left[P^{2}(Y\geq
Y^{\prime}|X)\right],$

and

$\lim_{n\rightarrow\infty}\mathbb{E}\left[\frac{2n+1}{n^{2}+2n}\frac{3\sum_{i=1}^{n-1}|r_{i+1}-r_{i}|}{n-1}\right]=6-12\mathbb{E}_{Y^{\prime}}\mathbb{E}_{X}\left[P^{2}(Y\geq
Y^{\prime}|X)\right],$

where $Y^{\prime}$ represents an independent copy of $Y$. Suppose
$X_{(m)}<x\leq X_{(m+1)}$, an important observation is

$\Delta_{n}=\frac{3}{n-1}\left[\sum_{i\neq
m}\mathbf{1}\\{(Y_{(i)}-y)(Y_{(i+1)}-y)<0\\}+|r^{*}_{m+2}-r^{*}_{m+1}|+|r^{*}_{m+1}-r^{*}_{m}|-|r_{m+1}-r_{m}|\right],$

where $r^{*}_{m+1}$ is the rank of $y$.

Let

$\displaystyle\Delta_{n1}$
$\displaystyle=\frac{3}{n-1}\left\\{\sum_{i=1}^{n}\mathbf{1}[(Y_{i}-y)(Y_{N(i)}-y)<0]\right\\}$
$\displaystyle=\frac{3}{n-1}\left\\{n-\sum_{i=1}^{n}\mathbf{1}(Y_{i}<y)\mathbf{1}[Y_{N(i)}<y]-\sum_{i=1}^{n}\mathbf{1}(Y_{i}>y)\mathbf{1}[Y_{N(i)}>y]\right\\},$

again by Lemma (9.10) of [8], we have

$\displaystyle\lim_{n\rightarrow\infty}\mathbb{E}\\{\mathbf{1}(Y_{i}<y)\mathbf{1}[Y_{N(i)}<y]\\}$
$\displaystyle=\mathbb{E}_{X}[P^{2}(Y<y|X)],$
$\displaystyle\lim_{n\rightarrow\infty}\mathbb{E}\\{\mathbf{1}(Y_{i}>y)\mathbf{1}[Y_{N(i)}>y]\\}$
$\displaystyle=\mathbb{E}_{X}[P^{2}(Y>y|X)],$

therefore

$\displaystyle\lim_{n\rightarrow\infty}\mathbb{E}(\Delta_{n1})$
$\displaystyle=6\mathbb{E}_{X}\left\\{P(Y>y|X)[1-P(Y>y|X)]\right\\}$
$\displaystyle=6\mathbb{E}_{X}\left[P(Y>y|X)\right]-6\mathbb{E}_{X}\left[P^{2}(Y>y|X)\right]$
$\displaystyle=6P(Y>y)-6\mathbb{E}_{X}\left[P^{2}(Y>y|X)\right].$

Next, we have

$\displaystyle\frac{3}{n-1}|r_{m+1}-r_{m}|$
$\displaystyle=\frac{3n}{n-1}\frac{1}{n}|r_{m+1}-r_{m}|$
$\displaystyle=\frac{3n}{n-1}|F_{Yn}(Y_{(m+1)})-F_{Yn}(Y_{(m)})|.$

Similar to $\Delta_{n1}$, we have

$\lim_{n\rightarrow\infty}\mathbb{E}\left[\frac{3}{n-1}|r_{m+1}-r_{m}|\right]=\lim_{n\rightarrow\infty}\mathbb{E}\left[3|F_{Y}(Y_{(m+1)})-F_{Y}(Y_{(m)})|\right],$
$\displaystyle\lim_{n\rightarrow\infty}\mathbb{E}\left\\{\min[F_{Y}(Y_{(m+1)}),F_{Y}(Y_{(m)})]\right\\}$
$\displaystyle=\lim_{n\rightarrow\infty}\mathbb{E}_{Y^{\prime}}\left[P(Y^{\prime}<Y|X=X_{(m)})P(Y^{\prime}<Y|X=X_{(m+1)})\right]$
$\displaystyle=\mathbb{E}_{Y^{\prime}}\left[P^{2}(Y^{\prime}<Y|X=x)\right],$

and

$\displaystyle\lim_{n\rightarrow\infty}\mathbb{E}\left[F_{Y}(Y_{(m)})\right]$
$\displaystyle=\lim_{n\rightarrow\infty}\mathbb{E}_{Y^{\prime}}\left[P(Y^{\prime}<Y|X=X_{(m)})\right]$
$\displaystyle=\mathbb{E}_{Y^{\prime}}\left[P(Y^{\prime}<Y|X=x)\right].$

Furthermore, we have
$\lim_{n\rightarrow\infty}\mathbb{E}\left[F_{Y}(Y_{(m+1)})\right]=\mathbb{E}_{Y^{\prime}}\left[P(Y^{\prime}<Y|X=x)\right]$
and

$\displaystyle\lim_{n\rightarrow\infty}\mathbb{E}(\Delta_{n2})$
$\displaystyle=\lim_{n\rightarrow\infty}\mathbb{E}\left[3|F_{Y}(Y_{(m+1)})-F_{Y}(Y_{(m)})|\right]$
$\displaystyle=\lim_{n\rightarrow\infty}3\mathbb{E}\left\\{F_{Y}(Y_{(m+1)})+F_{Y}(Y_{(m)})-2\min[F_{Y}(Y_{(m+1)}),F_{Y}(Y_{(m)})]\right\\}$
$\displaystyle=6\mathbb{E}_{Y^{\prime}}\left[P(Y^{\prime}<Y|X=x)\right]-6\mathbb{E}_{Y^{\prime}}\left[P^{2}(Y^{\prime}<Y|X=x)\right]$

Similarly, for $\Delta_{n3}$, we have

$\displaystyle\lim_{n\rightarrow\infty}\mathbb{E}(\Delta_{n3})$
$\displaystyle=6\mathbb{E}_{Y^{\prime}}\left[P(Y^{\prime}<Y|X=x)\right]+6\mathbb{E}_{Y^{\prime}}\left[P(Y^{\prime}<y)\right]-12\mathbb{E}_{Y^{\prime}}\left[P(Y^{\prime}<Y,Y^{\prime}<y|X=x)\right]$
$\displaystyle=6\mathbb{E}_{Y^{\prime}}\left[P(Y^{\prime}<Y|X=x)\right]+6P(Y<y)-12\mathbb{E}_{Y^{\prime}}\left[P(Y^{\prime}<Y,Y^{\prime}<y|X=x)\right]$

Summarizing the results above, we have

$\displaystyle\mbox{AESF}[(x,~{}y),~{}\xi,~{}F]=$
$\displaystyle-12\mathbb{E}_{Y^{\prime}}\mathbb{E}_{X}\left[P^{2}(Y^{\prime}<Y|X)\right]+6\mathbb{E}_{X}\left[P^{2}(Y>y|X)\right]-6\mathbb{E}_{Y^{\prime}}\left[P^{2}(Y^{\prime}<Y|X=x)\right]$
$\displaystyle+12\mathbb{E}_{Y^{\prime}}\left[P(Y^{\prime}<Y,Y^{\prime}<y|X=x)\right],$

which is uniformly bounded.

As illustrative examples, we present in Figure 4 below the $AESF$ of
Chatterjee’s correlation under three correlative patterns: (A) linear (B)
quadratic and (C) sinusoid. Let $Z\sim N(0,1)$ and $Z\perp X$, the three
settings are constructed as follows

  * A.

$X\sim N(0,1)$, $Y=0.7X+\sqrt{1-0.7^{2}}Z$.

  * B.

$X\sim U(-10,10)$, $Y=X^{2}+\sqrt{10}Z$.

  * C.

$X\sim U(-1,1)$, $Y=\cos(2\pi X)+0.5Z$.

[Figure 4 about here]

##  4 Discussion and conclusions

In this paper, we introduced a new robustness measure, $AESF$, which is
defined as the limit of the expected sensitivity function. The $AESF$ bridges
the gap between sensitivity and influence functions, exhibiting convenient
properties for both. Notably, it shares equivalence with influence functions
for many common estimators such as mean, variance, and Spearman’s correlation
and Kendall’s correlation. The $AESF$ possesses several key advantages. First,
unlike the sensitivity function which is generally random even when sample
size approaches infinity, the $AESF$ is always deterministic, which is a
desired property for robustness measures. Second, compared to the influence
function based on Gâteaux derivative, the $AESF$ are often more mathematically
manageable, particularly for complex statistical functionals. We illustrate
this with Chatterjee’s correlation and Dette-Siburg-Stoimenov’s dependence
measure as an example. Furthermore, one may use this new metric to redefine
some robustness concepts such as gross-error sensitivity and B-robustness,
further enriching the understanding of statistical stability.

While Theorems 1 and Corollary 1 establish sufficient conditions for
equivalence between $SF$, $AESF$, and $IF$ for a broad class of functionals
and estimators, a deeper understanding requires a sufficient and necessary
condition. The intricate relationship between $IF$ and $SF$ has been
extensively studied. Boos & Serfling (1980) demonstrated that under Fréchet
differentiability, $SF$ consistently estimates $IF$ asymptotically with
normality. Subsequently, Cuevas & Romo (1995) established the asymptotic
validity of bootstrap confidence bands for $SF$ as an estimator of $IF$ using
Gill’s generalized method for Hadamard differentiable operators. However,
these results hinge on pre-defined sufficient conditions. Achieving a
sufficient and necessary condition presents a significant theoretical
challenge, hence we leave it as an open question for future investigation.

The proposed robustness metric has far-reaching potential, in addition to the
nonparametric correlation estimators showcased in this work. For instance, it
readily applies to the robustness of diverse statistical and machine learning
models, even without requiring the explicit form of their underlying
functional. This versatility grants $AESF$ broad applicability within the
realm of model robustness assessment.

## Acknowledgement

The work was supported by an NSF DBI Biology Integration Institute (BII) grant
(award no. 2119968; PI-Ceballos).

## Appendix: Proof of Theorem 1

###### Proof.

We first derive $\mbox{IF}(x,~{}R,~{}F)$. Let $E_{g}=\int g(t)dF(t)$, we have

$\displaystyle\mbox{IF}(x,~{}R,~{}F)$ $\displaystyle=\lim_{\epsilon\rightarrow
0}\frac{\phi[\epsilon g(x)+(1-\epsilon)E_{g}]-\phi[E_{g}]}{\epsilon}$
$\displaystyle=[g(x)-E_{g}]\lim_{\epsilon\rightarrow
0}\frac{\phi\\{E_{g}+\epsilon[g(x)-E_{g}]\\}-\phi[E_{g}]}{\epsilon[g(x)-E_{g}]}$
$\displaystyle=[g(x)-E_{g}]\phi^{{}^{\prime}}[E_{g}].$

Next we show for any $x$,
$\mbox{SF}(x,~{}R_{n},~{}F_{n})\xrightarrow[]{a.s.}[g(x)-E_{g}]\phi^{{}^{\prime}}[E_{g}]$.
Let $(X_{1},~{}...,~{}X_{n})$ be i.i.d. samples, first we have

$\displaystyle\mbox{SF}(x,~{}R_{n},~{}F_{n})$
$\displaystyle=\frac{\phi\left\\{\frac{1}{n+1}\left[g(x)+\sum_{i=1}^{n}g(X_{i})\right]\right\\}-\phi\left[\frac{1}{n}\sum_{i=1}^{n}g(X_{i})\right]}{\frac{1}{n+1}}$
$\displaystyle=\left[g(x)-\frac{1}{n}\sum_{i=1}^{n}g(X_{i})\right]\frac{\phi\left\\{\frac{1}{n}\sum_{i=1}^{n}g(X_{i})+\frac{1}{n+1}\left[g(x)-\frac{1}{n}\sum_{i=1}^{n}g(X_{i})\right]\right\\}-\phi\left[\frac{1}{n}\sum_{i=1}^{n}g(X_{i})\right]}{\frac{1}{n+1}\left[g(x)-\frac{1}{n}\sum_{i=1}^{n}g(X_{i})\right]}.$

By the continuity and boundedness of $\phi^{{}^{\prime}}$, for any $\delta>0$,
there exists $\eta>0$, such that
$|\phi^{{}^{\prime}}(t)-\phi^{{}^{\prime}}(E_{g})|<\delta$ for
$E_{g}-\eta<t<E_{g}+\eta$. By SLLN, we have

$g(x)-\frac{1}{n}\sum_{i=1}^{n}g(X_{i})\xrightarrow[]{a.s.}g(x)-E_{g}$

and

$\frac{1}{n+1}\left[g(x)-\frac{1}{n}\sum_{i=1}^{n}g(X_{i})\right]\xrightarrow[]{a.s.}0.$

Therefore there exists $N_{1}$, such that for $n>N_{1}$,
$|\sum_{i=1}^{n}g(X_{i})/n-E_{g}|<\eta/2$ almost surely. Also, there exists
$N_{2}$, such that $n>N_{2}$,
$|\frac{1}{n+1}\left[g(x)-\frac{1}{n}\sum_{i=1}^{n}g(X_{i})\right]|<\eta/2$
almost surely. Thus by the continuity and boundedness of $\phi^{{}^{\prime}}$,
for $n>\max\\{N_{1},N_{2}\\}$, we have

$\left|\frac{1}{n}\sum_{i=1}^{n}g(X_{i})+\frac{1}{n+1}\left[g(x)-\frac{1}{n}\sum_{i=1}^{n}g(X_{i})-E_{g}\right]\right|<\eta,~{}a.s.$

and

$\left|\frac{\phi\left\\{\frac{1}{n}\sum_{i=1}^{n}g(X_{i})+\frac{1}{n+1}\left[g(x)-\frac{1}{n}\sum_{i=1}^{n}g(X_{i})\right]\right\\}-\phi\left[\frac{1}{n}\sum_{i=1}^{n}g(X_{i})\right]}{\frac{1}{n+1}\left[g(x)-\frac{1}{n}\sum_{i=1}^{n}g(X_{i})\right]}-\phi^{{}^{\prime}}(E_{g})\right|<\delta,~{}a.s.$

Therefore for any $x$,
$\mbox{SF}(x,~{}R_{n},~{}F_{n})\xrightarrow[]{a.s.}[g(x)-E_{g}]\phi^{{}^{\prime}}[E_{g}]$.

Finally, we show that
$\mbox{AESF}(x,~{}R,~{}F)=[g(x)-E_{g}]\phi^{{}^{\prime}}[E_{g}]$. Recall
$\mbox{AESF}(x,~{}R,~{}F)=\lim_{n\rightarrow\infty}\mathbb{E}[\mbox{SF}(x,~{}R_{n},~{}F_{n})]$.
By the boundedness of $\phi^{{}^{\prime}}$, there exists $M>0$, such that
$|\mbox{SF}(x,~{}R_{n},~{}F_{n})|<M\left|g(x)-\frac{1}{n}\sum_{i=1}^{n}g(X_{i})\right|$.
As
$\mathbb{E}\left|g(x)-\frac{1}{n}\sum_{i=1}^{n}g(X_{i})\right|\leq|g(x)|+\int|g(t)|dF(t)<\infty$,
by Lebesgue’s Dominated Convergence Theorem, we have
$\mbox{AESF}(x,~{}R,~{}F)=\mathbb{E}[\lim_{n\rightarrow\infty}\mbox{SF}(x,~{}R_{n},~{}F_{n})]$.
Since $\mbox{SF}(x,~{}R_{n},~{}F_{n})]$ is uniformly bounded, the almost sure
convergence also implies convergence in expectation, thus
$\mathbb{E}[\lim_{n\rightarrow\infty}\mbox{SF}(x,~{}R_{n},~{}F_{n})]=[g(x)-E_{g}]\phi^{{}^{\prime}}[E_{g}].$
This completes the proof. ∎

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## Figures

Figure 1: The $AESF$ of Kendall’s correlation for bivariate normal
distribution with $\rho=0.7$.

Figure 2: The $AESF$ of Spearman’s correlation for bivariate normal
distribution with $\rho=0.7$.

Figure 3: Comparison of Kendall’s and Spearman’s correlations in $AESF$, where
red highlights areas where Kendall’s correlation is more robust (smaller
absolute $AESF$), while blue indicate regions where Spearman’s correlation is
more robust.

Figure 4: The $AESF$ of Chatterjee’s correlation for (A) linear (B) quadratic
and (C) sinusoid, under Gaussian random noise.