# Quantifying and estimating dependence via sensitivity of conditional
distributions

Jonathan Ansari  Department of Artificial Intelligence and Human Interfaces,
University of Salzburg, Austria<EMAIL_ADDRESS>Patrick B.
Langthaler  Department of Artificial Intelligence and Human Interfaces,
University of Salzburg, Austria  Department of Neurology, Christian Doppler
Klinik, Paracelsus Medical University, Salzburg, Austria
<EMAIL_ADDRESS>Sebastian Fuchs  Department of
Artificial Intelligence and Human Interfaces, University of Salzburg, Austria
<EMAIL_ADDRESS>Wolfgang Trutschnig  Department of Artificial
Intelligence and Human Interfaces, University of Salzburg, Austria
<EMAIL_ADDRESS>
###### Abstract

Recently established, directed dependence measures for pairs $(X,Y)$ of random
variables build upon the natural idea of comparing the conditional
distributions of $Y$ given $X=x$ with the marginal distribution of $Y$. They
assign pairs $(X,Y)$ values in $[0,1]$, the value is $0$ if and only if $X,Y$
are independent, and it is $1$ exclusively for $Y$ being a function of $X$.
Here we show that comparing randomly drawn conditional distributions with each
other instead or, equivalently, analyzing how sensitive the conditional
distribution of $Y$ given $X=x$ is on $x$, opens the door to constructing
novel families of dependence measures $\Lambda_{\varphi}$ induced by general
convex functions $\varphi:\mathbb{R}\rightarrow\mathbb{R}$, containing, e.g.,
Chatterjee’s coefficient of correlation as special case. After establishing
additional useful properties of $\Lambda_{\varphi}$ we focus on continuous
$(X,Y)$, translate $\Lambda_{\varphi}$ to the copula setting, consider the
$L^{p}$-version and establish an estimator which is strongly consistent in
full generality. A real data example and a simulation study illustrate the
chosen approach and the performance of the estimator. Complementing the afore-
mentioned results, we show how a slight modification of the construction
underlying $\Lambda_{\varphi}$ can be used to define new measures of
explainability generalizing the fraction of explained variance.

###### keywords:

dependence measure, sensitivity, conditional distribution, Chatterjee’s
correlation coefficient, explainability, copula

##  1 Introduction

In classical statistics comparing two groups/populations is usually done by
considering specific parameters like the mean or the variance of the two
distributions $\mathbb{P}^{Y|X=x_{1}}$ and $\mathbb{P}^{Y|X=x_{2}}\,$. In
other words, one studies expressions of the form $\Delta(r(x_{1}),r(x_{2}))$
or $\Delta(v(x_{1}),v(x_{2}))$ for some non-negative functional/distance
$\Delta\,,$ where $r(x)=\mathbb{E}[Y|X=x]$ denotes the regression function
(conditional expectation) and $v(x)=\mathbb{V}(Y|X=x)$ the conditional
variance of $Y$ given $X=x$, respectively. Moving from the two-groups setting
(modelled by a binary random variable $X$) to the general case of an arbitrary
conditioning variable $X\,,$ it seems natural to examine how much the
conditional distribution of $Y$ given $X$ changes if $X$ changes, i.e., how
sensitive the conditional distribution $\mathbb{P}^{Y|X=x}$ of $Y$ given $X=x$
is on $x$. Intuitively, knowing the extent of sensitivity should provide
information on how much information on $Y$ we gain by knowing $X$. Taking into
account the distribution of the conditioning variable $X$, one natural
approach for quantifying sensitivity is to consider the average/expected value
of $\Delta(\mathbb{P}^{Y|X=x_{1}},\mathbb{P}^{Y|X=x_{2}})$ for a sample
$x_{1},x_{2}$ of $X$. Writing $\mathbb{P}^{X}\otimes\mathbb{P}^{X}$ for the
product measure of $\mathbb{P}^{X}$ with itself one could therefore consider
functionals of the form

$\Lambda_{\Delta}(Y|X):=\alpha_{\Delta}^{-1}\,\int_{\mathbb{R}^{2}}\Delta(\mathbb{P}^{Y|X=x_{1}},\mathbb{P}^{Y|X=x_{2}})\mathrm{\,d}(\mathbb{P}^{X}\otimes\mathbb{P}^{X})(x_{1},x_{2}),$
(1)

for some normalising constant $\alpha_{\Delta}>0\,.$ As a matter of fact,
various well-known measures can be represented in the form (1), including the
portion of explained variance or Chatterjee’s famous coefficient of rank
correlation, see Section 2. Comparing conditional distributions is certainly
not a new idea, see, e.g., [11, 12, 31] in the context of differential
privacy. However, to the best of our knowledge, the (recently fast growing)
literature on dependence measures (also known as measures of predictability,
i.e., measures attaining values in [0,1] and being minimal/maximal exclusively
for the case of independence/perfect dependence) mainly focuses on comparing
conditional and unconditional distributions, so on expressions of the form

$\displaystyle\int_{\mathbb{R}}\Delta(\mathbb{P}^{Y|X=x},\mathbb{P}^{Y})\mathrm{\,d}\mathbb{P}^{X}(x)\,,$
(2)

where $\Delta$ refers either to a $L^{2}$-distance (see, e.g., [2, 4, 6, 13,
15, 34, 36, 9]), the $L^{1}$-distance (see, e.g., [18, 22, 37]), some optimal
transport cost function (see, e.g., [29]), the maximum mean discrepancy (see,
e.g., [20]), or the Wasserstein distance (see, e.g., [38]).
On the one hand, for $\Delta$ denoting the $L^{2}$-distance between univariate
distribution functions the quantities in (1) and (2) can be shown to coincide,
see Example 1 in the next section. On the other hand, in general these
expressions may differ, so considering functionals of the form
$\Lambda_{\Delta}$ leads to novel measures of association and, in particular,
to new dependence measures, which are the main focus of this contribution.

The remainder of this paper is organized as follows: Section 2 first shows
that Chatterjee’s dependence measure can be represented in the form
$\Lambda_{\Delta}$ and proposes a family of functionals
$\Delta=\Delta_{\varphi}$ based on convex functions
$\varphi:\mathbb{R}\rightarrow\mathbb{R}$. Before focusing on these measures
for the rest of the paper, we show that using the same construction but
replacing the conditional distributions by the regression functions leads to a
large class of so-called explainability measures which can be seen as
generalization of the well-known fraction of explained variance. Section 3
proves that the afore-mentioned functionals
$\Lambda_{\Delta_{\varphi}}=\Lambda_{\varphi}$ are in fact dependence
measures, studies further (invariance and order) properties, then focuses on
continuous $(X,Y)$, and translates the dependence measures to the copula
setting. Based on these results, in Section 4 a copula-based checkerboard
estimator is proposed and shown to be strongly consistent in full generality,
i.e., without any smoothness restrictions on the underlying bivariate
copulas/distributions. A simulation study illustrating the performance of the
estimator complemented by a read data example in Section 5 round off the
paper.

In what follows we always consider real-valued general random variables $X,Y$
defined on a common probability space $(\Omega,\mathcal{A},\mathbb{P})\,$. We
will refer to $X$ as the exogenous and to $Y$ as the endogenous variable and
assume that the latter is non-degenerated, i.e., $\mathbb{P}^{Y}$ is not a
one-point distribution. Furthermore we will write
$X^{\prime}\stackrel{{\scriptstyle\mathrm{d}}}{{=}}X$ if, and only if
$X^{\prime}$ and $X$ have the same distribution, i.e., if
$\mathbb{P}^{X^{\prime}}=\mathbb{P}^{X}$ holds.

##  2 General setting and motivating examples

Let $\mathcal{M}$ denote the class of probability measures on $\mathbb{R}\,.$
For studying the sensitivity of conditional distributions we consider
functionals $\Delta:\mathcal{M}\times\mathcal{M}\rightarrow[0,\infty)$ that
may satisfy several desirable and natural properties such as symmetry, i.e.,
$\Delta(\mu,\nu)=\Delta(\nu,\mu)$ for all $\mu,\nu\in\mathcal{M}\,,$ or
positive definiteness, i.e., $\Delta(\mu,\nu)\geq 0$ for all
$\mu,\nu\in\mathcal{M}$ with equality if and only if $\mu=\nu\,.$
Given a bivariate random vector $(X,Y)$ on
$(\Omega,\mathcal{A},\mathbb{P})\,,$ the mapping $\Lambda_{\Delta}(Y|X)$ is
defined according to (1), whereby we implicitly assume that the mapping
$(x_{1},x_{2})\mapsto\Delta\big{(}\mathbb{P}^{Y|X=x_{1}},\mathbb{P}^{Y|X=x_{2}}\big{)}$
is integrable. Moreoever, for the rest of this paper the normalising constant
$\alpha_{\Delta}$ is defined by

$\displaystyle\alpha_{\Delta}:=\sup\left\\{\int_{\mathbb{R}^{2}}\Delta(\mathbb{P}^{Y^{\prime}|X^{\prime}=x_{1}},\mathbb{P}^{Y^{\prime}|X^{\prime}=x_{2}})\mathrm{\,d}(\mathbb{P}^{X^{\prime}}\otimes\mathbb{P}^{X^{\prime}})(x_{1},x_{2}):\,(X^{\prime},Y^{\prime})\in\mathcal{R}(Y)\right\\}$
(3)

where $\mathcal{R}(Y)$ denotes the family of all random vectors
$(X^{\prime},Y^{\prime})\colon\Omega\to\mathbb{R}^{2}$ with
$Y^{\prime}\stackrel{{\scriptstyle\mathrm{d}}}{{=}}Y$. Simple expressions for
$\alpha_{\Delta}$ will be derived later on. Whenever the supremum is positive
and finite we obviously have that $\Lambda_{\Delta}\in[0,1]$. By definition,
$\Lambda_{\Delta}$ assigns every bivariate random vector $(X,Y)$ a non-
negative number which depends only on the distribution of $(X,Y).$

In the sequel we study properties of the functional $\Lambda_{\Delta}$ for
several classes of $\Delta$ and characterize in particular the maximal
elements determining the value of the constant $\alpha_{\Delta}\,.$ Notice
that for $\Delta$ being positive definite obviously $\Lambda_{\Delta}$
characterizes independence since independence of $X$ and $Y$ means that the
conditional distributions $(\mathbb{P}^{Y|X=x})_{x\in\mathbb{R}}$ do not
depend on $x$, which for such $\Delta$ is equivalent to
$\Lambda_{\Delta}(Y|X)=0$.

######  Remark 2.1.

Considering the special case of $\Delta$ being a metric on $\mathcal{M}$, the
functional $\Lambda_{\Delta}(Y|X)$ boils down to the average $\Delta$-distance
of two randomly (according to $\mathbb{P}^{X}$) drawn conditional
distributions $\mathbb{P}^{Y|X=x_{1}},\mathbb{P}^{Y|X=x_{2}}$. If, for
example, $\Delta$ is the Wasserstein distance $\mathcal{W},$ then the
functional in (1) is given by
$\Lambda_{\Delta}(Y|X)=\alpha_{\Delta}^{-1}\int_{\mathbb{R}^{2}}\mathcal{W}(\mathbb{P}^{Y|X=x_{1}},\mathbb{P}^{Y|X=x_{2}})\mathrm{\,d}\mathbb{P}^{X}\otimes\mathbb{P}^{X}(x_{1},x_{2})$,
see [38] for measures of association with Wasserstein distances of the form
(2). For a discussion of the case where $\Delta$ corresponds to
$L^{p}$-metrics we refer to Section 3.3.

The subsequent two examples illustrate the broadness of the
$\Lambda_{\Delta}$-approach according to (1) in the context of quantifying
(directed) dependence as well as explainability in terms of the sensitivity of
conditional distributions. They show that well-known statistical concepts can
either be expressed in terms of (1) with adequately chosen $\Delta$.

######  Example 1 (Chatterjee’s coefficient of correlation).

Consider $\Delta$ given by

$\displaystyle\Delta\big{(}\mathbb{P}^{Y|X=x_{1}},\mathbb{P}^{Y|X=x_{2}}\big{)}$
$\displaystyle:=\int_{\mathbb{R}}\Big{(}\mathbb{P}(Y\geq
y|X=x_{1})-\mathbb{P}(Y\geq
y|X=x_{2})\Big{)}^{2}\mathrm{\,d}\mathbb{P}^{Y}(y)\,.$

Then $\Lambda_{\Delta}$ defined by (1) coincides with Chatterjee’s famous
coefficient of correlation (see [6, 9]) since using change of coordinates we
have

$\displaystyle\Lambda_{\Delta}(Y|X)$ $\displaystyle=$
$\displaystyle\alpha_{\Delta}^{-1}\,\int_{\mathbb{R}^{2}}\int_{\mathbb{R}}\Big{(}\mathbb{P}(Y\geq
y|X=x_{1})-\mathbb{P}(Y\geq
y|X=x_{2})\Big{)}^{2}\mathrm{\,d}\mathbb{P}^{Y}(y)\mathrm{\,d}(\mathbb{P}^{X}\otimes\mathbb{P}^{X})(x_{1},x_{2})$
$\displaystyle=$ $\displaystyle
2\,\alpha_{\Delta}^{-1}\,\int_{\mathbb{R}}\left(\int_{\mathbb{R}}\mathbb{P}(Y\geq
y|X=x)^{2}\mathrm{\,d}\mathbb{P}^{X}(x)-\left(\int_{\mathbb{R}}\mathbb{P}(Y\geq
y|X=x)\mathrm{\,d}\mathbb{P}^{X}(x)\right)^{2}\right)\mathrm{\,d}\mathbb{P}^{Y}(y)$
$\displaystyle=$
$\displaystyle\frac{\int_{\mathbb{R}}\mathbb{V}(\mathbb{P}(Y\geq
y|X))\,\mathrm{\,d}\mathbb{P}^{Y}(y)}{\int_{\mathbb{R}}\mathbb{V}(\mathds{1}_{\\{Y\geq
y\\}})\mathrm{\,d}\mathbb{P}^{Y}(y)}.$

Thereby according to the afore-mentioned references again using change of
coordinates and Fubini’s theorem, and letting $Y_{1},Y_{2},Y$ be i.i.d., the
normalizing constant $\alpha_{\Delta}$ simplifies to (also see Theorem 3.2)

$\displaystyle\begin{split}\alpha_{\Delta}&=\left(2\,\int_{\mathbb{R}}\mathbb{V}(\mathds{1}_{\\{Y\geq
y\\}})\mathrm{\,d}\mathbb{P}^{Y}(y)\right)\\\
&=\left(\int_{\mathbb{R}}\int_{\Omega}\Big{(}\mathds{1}_{[y,\infty)}(Y_{1}(\omega))-\mathbb{P}(Y_{1}\geq
y)\Big{)}^{2}+\Big{(}\mathds{1}_{[y,\infty)}(Y_{2}(\omega))-\mathbb{P}(Y_{2}\geq
y)\Big{)}^{2}\mathrm{\,d}\mathbb{P}(\omega)\mathrm{\,d}\mathbb{P}^{Y}(y)\right)\\\
&=\left(\int_{\mathbb{R}}\int_{\Omega}\Big{(}\mathds{1}_{[y,\infty)}(Y_{1}(\omega))-\mathbb{P}(Y_{1}\geq
y)-\mathds{1}_{[y,\infty)}(Y_{2}(\omega))+\mathbb{P}(Y_{2}\geq
y)\Big{)}^{2}\mathrm{\,d}\mathbb{P}(\omega)\mathrm{\,d}\mathbb{P}^{Y}(y)\right)\\\
&=\left(\int_{\mathbb{R}^{2}}\int_{\mathbb{R}}\Big{(}\mathds{1}_{[y,\infty)}(y_{1})-\mathds{1}_{[y,\infty)}(y_{2})\Big{)}^{2}\mathrm{\,d}\mathbb{P}^{Y}(y)\mathrm{\,d}(\mathbb{P}^{Y}\otimes\mathbb{P}^{Y})(y_{1},y_{2})\right)\\\
&=\int_{\mathbb{R}^{2}}\Delta\left(\delta_{y_{1}},\delta_{y_{2}}\right)\mathrm{\,d}\mathbb{P}^{Y}\otimes\mathbb{P}^{Y}(y_{1},y_{2})\,,\end{split}$
(4)

and where $\delta_{x}$ denotes the Dirac measure at $x\in\mathbb{R}\,.$ Hence,
by the properties of Chatterjee’s coefficient of correlation, the above choice
of $\Delta$ yields that the functional $\Lambda_{\Delta}$ is a measure of
dependence, i.e., $\Lambda_{\Delta}(Y|X)\in[0,1]$, $\Lambda_{\Delta}(Y|X)=0$
if and only if $X$ and $Y$ are independent, and $\Lambda_{\Delta}(Y|X)=1$
exclusively if $Y$ is completely dependent on $X\,,$ i.e., there exists some
measurable function $f\colon\mathbb{R}\to\mathbb{R}$ with $Y=f(X)$ almost
surely.

Motivated by the above example, Section 3 introduces and investigates a large
family of functionals $\Lambda_{\Delta}$, where $\Delta$ depends only on the
difference of conditional distribution functions rescaled by some measurable
function $\varphi\colon\mathbb{R}\to\mathbb{R}$:

$\displaystyle\Delta_{\varphi}\big{(}\mathbb{P}^{Y|X=x_{1}},\mathbb{P}^{Y|X=x_{2}}\big{)}$
$\displaystyle:=\int_{\mathbb{R}}\varphi\left(F_{Y|X=x_{1}}(y)-F_{Y|X=x_{2}}(y)\right)\mathrm{\,d}\mathbb{P}^{Y}(y)$
(5)

We will show in Theorem 3.2 that for $\varphi$ being convex and strictly
convex in $0=\varphi(0)$ the functional
$\Lambda_{\Delta_{\varphi}}=:\Lambda_{\varphi}$ is a dependence measures (with
normalizing constant $\alpha_{\Delta_{\varphi}}$ according to (4)).

######  Example 2 (Cramér-von-Mises indices; fraction of explained variance).

Suppose that $Y$ is square-integrable. Denote by $r(x):=\mathbb{E}(Y|X=x)$ the
regression function of $Y$ given $X=x$, and consider

$\displaystyle\Delta\big{(}\mathbb{P}^{Y|X=x_{1}},\mathbb{P}^{Y|X=x_{2}}\big{)}$
$\displaystyle:=$
$\displaystyle\left(\mathbb{E}(Y|X=x_{1})-\mathbb{E}(Y|X=x_{2})\right)^{2}=(r(x_{1})-r(x_{2}))^{2}\,.$

Then the normalizing constant $\alpha_{\Delta}$ is given by

$\displaystyle\alpha_{\Delta}$
$\displaystyle=\left(\int_{\mathbb{R}^{2}}\left(y_{1}-y_{2}\right)^{2}\mathrm{\,d}(\mathbb{P}^{Y}\otimes\mathbb{P}^{Y})(y_{1},y_{2})\right)=2\,\mathbb{V}(Y)$

and the functional $\Lambda_{\Delta}(Y|X)$ in (1) coincides with the fraction
of explained variance of $Y$ given $X$, also known as Sobol index or Cramér-
von-Mises index (see [15]), i.e.,

$\displaystyle\Lambda_{\Delta}(Y|X)$ $\displaystyle=$
$\displaystyle\alpha_{\Delta}^{-1}\,\int_{\mathbb{R}^{2}}\left(\mathbb{E}(Y|X=x_{1})-\mathbb{E}(Y|X=x_{2})\right)^{2}\mathrm{\,d}(\mathbb{P}^{X}\otimes\mathbb{P}^{X})(x_{1},x_{2})$
$\displaystyle=$ $\displaystyle
2\,\alpha_{\Delta}^{-1}\,\left(\int_{\mathbb{R}}r^{2}(x)\mathrm{\,d}\mathbb{P}^{X}(x)-\left(\int_{\mathbb{R}}r(x)\mathrm{\,d}\mathbb{P}^{X}(x)\right)^{2}\right)$
$\displaystyle=$ $\displaystyle
2\,\alpha_{\Delta}^{-1}\,\left(\mathbb{E}((r\circ X)^{2})-(\mathbb{E}(r\circ
X))^{2}\right)$ $\displaystyle=$ $\displaystyle
2\,\alpha_{\Delta}^{-1}\,\mathbb{V}(r\circ
X)=\frac{\mathbb{V}(\mathbb{E}(Y|X))}{\mathbb{V}(Y)}=\frac{\int_{\mathbb{R}}\left(\mathbb{E}(Y|X=x)-\mathbb{E}(Y)\right)^{2}\mathrm{\,d}\mathbb{P}^{X}(x)}{\mathbb{V}(Y)}\,.$

Recall that in this case $\Lambda_{\Delta}(Y|X)\in[0,1]$ and we have
$\Lambda_{\Delta}(Y|X)=0$ if, and only if $\mathbb{V}(r\circ X)=0$ (which is
the case if $X$ and $Y$ are independent but not vice versa), and
$\Lambda_{\Delta}(Y|X)=1$ if, and only if $Y$ is completely dependent on $X$.
In the case of an affine regression function $r(x)=ax+b$, the quantity
$\Lambda_{\Delta}(Y|X)$ then coincides with the squared Pearson correlation
$\rho$ of $X$ and $Y$, i.e.

$\Lambda_{\Delta}(Y|X)=\frac{(Cov(X,Y))^{2}}{\mathbb{V}(X)\mathbb{V}(Y)}=\rho^{2}(X,Y)\,.$

As demonstrated in Example 2, $\Lambda_{\Delta}(Y|X)$ coincides with the
$L^{2}$-Cramér-von-Mises index, i.e., it coincides with the expected squared
distance between the conditional and the unconditional expectation. This
motivates to investigate another subclass of functionals $\Lambda_{\Delta}$
where $\Delta$ now depends only on the difference of conditional expectations,
weighted again by some measurable function
$\psi\colon\mathbb{R}\to\mathbb{R}$, i.e.,

$\displaystyle\Delta\big{(}\mathbb{P}^{Y|X=x_{1}},\mathbb{P}^{Y|X=x_{2}}\big{)}$
$\displaystyle:=\psi\left(\mathbb{E}(Y|X=x_{1})-\mathbb{E}(Y|X=x_{2})\right).$
(6)

This leads to a novel class of Cramer-von-Mises indices solely based on
sensitivity of the conditional expectations. For some specific choices of
$\psi$ in (6), the associated functional $\Lambda_{\Delta}$ in (1) relates the
variability of the conditional distribution to the variability of the
unconditional distribution as in (2), see, e.g., [14] for several Gini-type
measures of risk and variability.
For the special case where $\Delta=\Delta_{\psi}$ is of the form (6), the
functional $\Lambda_{\Delta_{\psi}}=:\Delta_{\psi}$ is given by

$\displaystyle\Lambda_{\psi}(Y|X):$
$\displaystyle=\alpha_{\psi}^{-1}\int_{\mathbb{R}^{2}}\psi\left(\mathbb{E}(Y|X=x_{1})-\mathbb{E}(Y|X=x_{2})\right)\mathrm{\,d}(\mathbb{P}^{X}\otimes\mathbb{P}^{X})(x_{1},x_{2})$
(7)

with normalizing constant $\alpha_{\psi}$ fulfilling

$\alpha_{\psi}=\sup\left\\{\int_{\mathbb{R}^{2}}\psi\left(\mathbb{E}(Y^{\prime}|X^{\prime}=x_{1})-\mathbb{E}(Y^{\prime}|X^{\prime}=x_{2})\right)\mathrm{\,d}(\mathbb{P}^{X^{\prime}}\otimes\mathbb{P}^{X^{\prime}})(x_{1},x_{2}):(X^{\prime},Y^{\prime})\in\mathcal{R}(Y)\right\\}\,,$

whenever the supremum is finite and positive.

In order to assure that $\Lambda_{\psi}$ according to (7) is a measure of
explainability the following property will be key: A convex function
$\phi\colon\mathbb{R}\to\mathbb{R}$ is said to be strictly convex in $0$ if
$(\phi(-\varepsilon)+\phi(\varepsilon))/2>\phi(0)$ holds for every
$\varepsilon>0\,.$ Notice that in the subsequent theorem $\psi$ is also
allowed to attain negative values.

######  Theorem 2.2 (Measures of explainability).

Suppose that $\psi(2Y)$ and $\psi(-2Y)$ are integrable and that
$\psi\colon\mathbb{R}\to\mathbb{R}$ is convex and strictly convex in $0$ with
$\psi(0)=0$. Then the constant $\alpha_{\psi}$ is given by

$\displaystyle\alpha_{\psi}=\int_{\mathbb{R}^{2}}\psi\left(y_{1}-y_{2}\right)\mathrm{\,d}(\mathbb{P}^{Y}\otimes\mathbb{P}^{Y})(y_{1},y_{2})\,.$
(8)

Furthermore, $\Lambda_{\psi}(Y|X)$ defined according to (7) is a measure of
explainability, i.e., it fulfills the following three properties:

  1. (i)

$\Lambda_{\psi}(Y|X)\in[0,1]\,.$

  2. (ii)

$\Lambda_{\psi}(Y|X)=0$ if, and only if $\mathbb{V}(\mathbb{E}(Y|X))=0$, i.e.,
$\mathbb{E}(Y|X)=\mathbb{E}[Y]$ almost surely.

  3. (iii)

$\Lambda_{\psi}(Y|X)=1$ if, and only if $Y$ is completely dependent on $X$,
i.e., $\mathbb{E}(Y|X)=Y$ almost surely.

###### Proof.

We first prove property (i) and eq. (8) and proceed as follows: Let $Y_{1}$
and $Y_{2}$ be independent copies of $Y$. Then using convexity of $\psi$ and
the integrability assumption the right-hand side of eq. (8) is finite since

$\displaystyle\mathbb{E}\left|\psi(Y_{1}-Y_{2})\right|$
$\displaystyle=\mathbb{E}\left|\psi\left(\frac{2Y_{1}+(-2Y_{2})}{2}\right)\right|$
$\displaystyle\leq\mathbb{E}\left|\psi(2Y_{1})+\psi(-2Y_{2})\right|\leq\mathbb{E}|\psi(2Y)|+\mathbb{E}|\psi(-2Y)|<\infty.$

If $Y=f(X)$ almost surely, then using change of coordinates it yields

$\displaystyle\begin{split}&\int_{\mathbb{R}^{2}}\psi\left(\mathbb{E}(Y|X=x_{1})-\mathbb{E}(Y|X=x_{2})\right)\mathrm{\,d}(\mathbb{P}^{X}\otimes\mathbb{P}^{X})(x_{1},x_{2})\\\
&=\int_{\mathbb{R}^{2}}\psi\left(f(x_{1})-f(x_{2})\right)\mathrm{\,d}(\mathbb{P}^{X}\otimes\mathbb{P}^{X})(x_{1},x_{2})\\\
&=\int_{\mathbb{R}^{2}}\psi\left(y_{1}-y_{2}\right)\mathrm{\,d}(\mathbb{P}^{f(X)}\otimes\mathbb{P}^{f(X)})(y_{1},y_{2})\\\
&=\int_{\mathbb{R}^{2}}\psi\left(y_{1}-y_{2}\right)\mathrm{\,d}(\mathbb{P}^{Y}\otimes\mathbb{P}^{Y})(y_{1},y_{2}).\end{split}$
(9)

Hence, using disintegration and Jensen’s inequality it follows for all
$(X^{\prime},Y^{\prime})\in\mathcal{R}(Y)$ that

$\displaystyle 0$
$\displaystyle=\psi(0)=\psi\left(\mathbb{E}(Y^{\prime})-\mathbb{E}(Y^{\prime})\right)$
$\displaystyle=\psi\left(\int_{\mathbb{R}^{2}}\mathbb{E}[Y^{\prime}|X^{\prime}=x_{1}]-\mathbb{E}[Y^{\prime}|X^{\prime}=x_{2}]\mathrm{\,d}(\mathbb{P}^{X^{\prime}}\otimes\mathbb{P}^{X^{\prime}})(x_{1},x_{2})\right)$
$\displaystyle\leq\int_{\mathbb{R}^{2}}\psi\left(\mathbb{E}[Y^{\prime}|X^{\prime}=x_{1}]-\mathbb{E}[Y^{\prime}|X^{\prime}=x_{2}]\right)\mathrm{\,d}(\mathbb{P}^{X^{\prime}}\otimes\mathbb{P}^{X^{\prime}})(x_{1},x_{2})$
(10)
$\displaystyle=\int_{\mathbb{R}^{2}}\psi\left(\int_{\mathbb{R}^{2}}y_{1}-y_{2}\mathrm{\,d}(\mathbb{P}^{Y^{\prime}|X^{\prime}=x_{1}}\otimes\mathbb{P}^{Y^{\prime}|X^{\prime}=x_{2}})(y_{1},y_{2})\right)\mathrm{\,d}(\mathbb{P}^{X^{\prime}}\otimes\mathbb{P}^{X^{\prime}})(x_{1},x_{2})$
$\displaystyle\leq\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\psi\left(y_{1}-y_{2}\right)\mathrm{\,d}(\mathbb{P}^{Y^{\prime}|X^{\prime}=x_{1}}\otimes\mathbb{P}^{Y^{\prime}|X^{\prime}=x_{2}})(y_{1},y_{2})\mathrm{\,d}(\mathbb{P}^{X^{\prime}}\otimes\mathbb{P}^{X^{\prime}})(x_{1},x_{2})$
$\displaystyle=\int_{\mathbb{R}^{2}}\psi\left(y_{1}-y_{2}\right)\mathrm{\,d}(\mathbb{P}^{Y^{\prime}}\otimes\mathbb{P}^{Y^{\prime}})(y_{1},y_{2})$
(11)
$\displaystyle\leq\sup\left\\{\int_{\mathbb{R}^{2}}\psi\left(\mathbb{E}(Y^{\prime}|X^{\prime}=x_{1})-\mathbb{E}(Y^{\prime}|X^{\prime}=x_{2})\right)\mathrm{\,d}(\mathbb{P}^{X^{\prime}}\otimes\mathbb{P}^{X^{\prime}})(x_{1},x_{2}):(X^{\prime},Y^{\prime})\in\mathcal{R}(Y)\right\\}$
$\displaystyle=\alpha_{\psi}.$

Using eq. (9) and applying eq. (10) to eq. (11) on the elements the supremum
is taken over shows that the third inequality becomes an equality. This proves
eq. (8) and implies $\Lambda_{\psi}(Y|X)\in[0,1]$.
(ii): Obviously $\mathbb{V}(\mathbb{E}(Y|X))=0$ if, and only if
$\mathbb{E}(Y|X=x_{1})=\mathbb{E}(Y)=\mathbb{E}(Y|X=x_{2})$ for
$\mathbb{P}^{X}\otimes\mathbb{P}^{X}$-almost all $x_{1},x_{2}\in\mathbb{R}$.
The latter, however, is equivalent to the integrand in (7) being zero almost
everywhere since $\psi$ is strictly convex in $0$ and $\psi(0)=0$.
(iii): If $Y$ is completely dependent on $X$, then (9) and (8) imply

$\displaystyle\alpha_{\psi}\,\Lambda_{\psi}(Y|X)$
$\displaystyle=\int_{\mathbb{R}^{2}}\psi\left(\mathbb{E}(Y|X=x_{1})-\mathbb{E}(Y|X=x_{2})\right)\mathrm{\,d}(\mathbb{P}^{X}\otimes\mathbb{P}^{X})(x_{1},x_{2})$
$\displaystyle=\int_{\mathbb{R}^{2}}\psi\left(y_{1}-y_{2}\right)\mathrm{\,d}(\mathbb{P}^{Y}\otimes\mathbb{P}^{Y})(y_{1},y_{2})=\alpha_{\psi}.$

Since $\mathbb{P}^{Y}$ is non-degenerate, complete dependence of $Y$ on $X$
yields $\mathbb{E}(Y|X)=Y$ and we do not have $Y=\mathbb{E}(Y)$ almost surely.
Hence, using (i) and (ii) the constant $\alpha_{\psi}$ is positive and
$\Lambda_{\psi}(Y|X)=1$ follows.
To show the reverse implication first recall that by Jensen’s inequality and
eq. (8) we have

$\displaystyle\begin{split}\alpha_{\psi}\Lambda_{\psi}(Y|X)&=\int_{\mathbb{R}^{2}}\psi\Big{(}\int_{\mathbb{R}^{2}}(y_{1}-y_{2})\mathrm{\,d}\mathbb{P}^{Y|X=x_{1}}\otimes\mathbb{P}^{Y|X=x_{2}}(y_{1},y_{2})\Big{)}\mathrm{\,d}\mathbb{P}^{X}\otimes\mathbb{P}^{X}(x_{1},x_{2})\\\
&\leq\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\psi(y_{1}-y_{2})\mathrm{\,d}\mathbb{P}^{Y|X=x_{1}}\otimes\mathbb{P}^{Y|X=x_{2}}\mathrm{\,d}\mathbb{P}^{X}\otimes\mathbb{P}^{X}(x_{1},x_{2})=\alpha_{\psi}.\end{split}$
(12)

For $\Lambda_{\psi}(Y|X)=1$ the above inequality becomes an equality there
exists some Borel set $G\subseteq\mathbb{R}^{2}$ with
$\mathbb{P}^{X}\otimes\mathbb{P}^{X}(G)=1$ such that for $(x_{1},x_{2})\in G$
and $Y_{i}\sim\mathbb{P}^{Y|X=x_{i}}\,,$ $i\in\\{1,2\\}\,,$ $Y_{1}$ and
$Y_{2}$ independent,

$\displaystyle\text{either}\quad Y_{1}\leq Y_{2}~{}\,\text{almost
surely}\quad\text{or}\quad Y_{1}\geq Y_{2}~{}\,\text{almost surely.}$ (13)

In fact, otherwise Jensen’s inequality would be strict due to strict convexity
of $\psi$ at $0=\psi(0)$, which contradicts $\Lambda_{\psi}(Y|X)=1$. Letting
$G_{x}:=\\{y\in\mathbb{R}\colon(x,y)\in G\\}$ denote the $x$-cut of $G$,
defining $W:=\\{x\in\mathbb{R}\colon\mathbb{P}^{X}(G_{x})=1\\}$ and using
disintegration obviously yields $\mathbb{P}^{X}(W)=1$. To simplify notation
let $l(x)$ and $u(x)$ denote the infimum and the supremum of the support of
$\mathbb{P}^{Y|X=x}\,,$ i.e.,

$\displaystyle
l(x):=\inf(\mathop{\mathrm{supp}}(\mathbb{P}^{Y|X=x}))\leq\sup(\mathop{\mathrm{supp}}(\mathbb{P}^{Y|X=x}))=:u(x).$
(14)

Then according to (13) for each $(x_{1},x_{2})\in G$ the open intervals fulfil
$(l(x_{1}),u(x_{1}))\cap(l(x_{2}),u(x_{2}))=\emptyset$. Finally define

$\displaystyle V:=\\{x\in\mathbb{R}\colon\mathbb{P}^{Y|X=x}\text{ is non-
degenerate}\\}=\\{x\in\mathbb{R}\colon\mathbb{V}(Y|X=x)>0\\}\,.$ (15)

For completing the proof it suffices to show

$\displaystyle\mathbb{P}^{X}(V)=0\,$ (16)

since it then follows that $\mathbb{P}^{Y|X=x}$ is degenerate for
$\mathbb{P}^{X}$-almost all $x\in\mathbb{R}\,,$ implying that $Y$ is
completely dependent on $X$.
Assume, on the contrary, that $\mathbb{P}^{X}(V)>0$ holds. Then considering
$\mathbb{P}^{X}(W)=1$ it follows that

$\displaystyle\mathbb{P}^{X}(V\cap W)=\mathbb{P}^{X}(V)>0\,.$ (17)

Let $x\in V\cap W$ be arbitrary but fixed. Then by contruction we have

$\displaystyle(l(x),u(x))\cap(l(z),u(z))=\emptyset$ (18)

for all $z\in\mathbb{R}$ with $(x,z)\in G$, so in particular for all $z\in
V\cap W\cap G_{x}$. Considering $\mathbb{P}^{X}(G_{x})=1=\mathbb{P}^{X}(W)$ it
follows that $0<\mathbb{P}^{X}(V)=\mathbb{P}^{X}(V\cap W\cap G_{x})$. Hence,
(18) holds in particular for $\mathbb{P}^{X}$-almost every $z\in V\cap W$.
Finally, let $q_{1},q_{2},q_{3},\ldots$ be an enumeration of the rational
numbers in $\mathbb{R}$. Then defining $M_{i}$ by $M_{i}:=\\{x\in V\cap
W\colon q_{i}\in(l(x),u(x))\\}$ yields $\bigcup_{i=1}^{\infty}M_{i}=V\cap W$
since the rationals are dense in $\mathbb{R}$. Using (17) as well as
sub-$\sigma$-additivity yields

$\displaystyle 0<\mathbb{P}^{X}(V\cap
W)=\mathbb{P}^{X}\Big{(}\bigcup_{i=1}^{\infty}M_{i}\Big{)}\leq\sum_{i=1}^{\infty}\mathbb{P}^{X}(M_{i}),$
(19)

so there exists some $i_{0}\in\mathbb{N}$ with $\mathbb{P}^{X}(M_{i_{0}})>0$.
The latter, however, contradicts (18), so $\mathbb{P}^{X}(V)>0$ can not hold,
and the proof is complete. ∎

######  Remark 2.3.

Our proof of assertion (iii) in Theorem 2.2 requires the assumption of strict
convexity of $\psi$ in $0$ and uses the fact that Jensen’s inequality applied
to the difference of two non-degenerate i.i.d. random variables is strict if
$\varphi$ is strictly convex in $0$. Hence, equality can only hold if the
underlying random variables are degenerate. Due to the integral with respect
to $\mathbb{P}^{X}\otimes\mathbb{P}^{X}\,,$ it has to be shown that Jensen’s
inequality holds for the inner integral
$\mathbb{P}^{X}\otimes\mathbb{P}^{X}$-almost surely. For $\mathbb{P}^{X}$
having at least one point mass the proof can be simplified significantly,
since then $\mathbb{P}^{X}(V)=0$ follows immediately.

##  3 A new class of dependence measures

For the rest of the paper we now focus on the functionals $\Lambda_{\varphi}$
and derive simple sufficient conditions for $\varphi$ in (5) assuring that
$\Lambda_{\Delta}(Y|X)$ defined by (1) is a dependence measure. Following the
recent literature we refer to $\kappa$ as a _(directed) dependence measure_
(frequently also referred to as measure of predictability) for bivariate
random vectors $(X,Y)$ if it satisfies the following axioms, see, e.g., [2, 5,
6, 8, 19, 35, 37] and the references therein:

  1. (A1)

$\kappa(Y|X)\in[0,1]$.

  2. (A2)

$\kappa(Y|X)=0$ if, and only if $Y$ and $X$ are independent.

  3. (A3)

$\kappa(Y|X)=1$ if and only if $Y$ is completely dependent on $X$, i.e., there
exists some measurable transformation $f:\mathbb{R}\to\mathbb{R}$ such that
$Y=f(X)$ holds almost surely.

Due to the axioms (A2) and (A3), the values $0$ and $1$ of a measure of
predictability $\kappa$ have a clear interpretation. The meaning of $\kappa\,$
attaining a specific value in the interval $(0,1)$, however, is not that
unequivocal - nevertheless, considering that the values in $[0,1]$ are
naturally ordered, they offer a clear comparative interpretation: the higher
the value of $\kappa$ the higher the extent of dependence of $Y$ on $X$. This
aspect motivates the study of dependence orderings $\prec$ that are compatible
with $\kappa$ in the following sense:

  1. (P1)

_Monotonicity_ : If $(X,Y)\prec(X^{\prime},Y^{\prime})$, then
$\kappa(Y|X)\leq\kappa(Y^{\prime}|X^{\prime})$.

It is further desirable - particularly in the context of estimation - to have
an understanding of the measures’ behavior in terms of convergence:

  1. (P2)

_Continuity_ : If $(X_{n},Y_{n})_{n\in\mathbb{N}}$ converges (in some sense)
to $(X,Y)$, then $\lim_{n\to\infty}\kappa(Y_{n}|X_{n})=\kappa(Y|X)$.

Moreover the following invariance properties seem desirable and will be
studied in the sequel:

  1. (P3)

_Invariance of $X$_: For bijective $h$ we have $\kappa(Y|X)=\kappa(Y|h(X))$.

  2. (P4)

_Invariance of $Y$_: If $g$ is strictly increasing, then
$\kappa(Y|X)=\kappa(g(Y)|X)$ holds.

  3. (P5)

_Version-Invariance_ : If
$(X,Y)\stackrel{{\scriptstyle\mathrm{d}}}{{=}}(X^{\prime},Y^{\prime})\,,$ then
$\kappa(X,Y)=\kappa(X,Y)\,.$

Note that property (P5) is also misleadingly referred to as law invariance in
the context of risk measures.

###  3.1 The class $\Lambda_{\varphi}$

For the special case where $\Delta$ is of the form (5) the functional
$\Lambda_{\Delta}$ is given by

$\displaystyle\Lambda_{\varphi}(Y|X):$
$\displaystyle=\alpha_{\varphi}^{-1}\int_{\mathbb{R}^{2}}\int_{\mathbb{R}}\varphi(F_{Y|X=x_{1}}(y)-F_{Y|X=x_{2}}(y))\mathrm{\,d}\mathbb{P}^{Y}(y)\mathrm{\,d}(\mathbb{P}^{X}\otimes\mathbb{P}^{X})(x_{1},x_{2})$
(20)

with normalizing constant $\alpha_{\varphi}$ according to (3) defined by

$\displaystyle\alpha_{\varphi}$
$\displaystyle:=\sup\left\\{\int_{\mathbb{R}^{2}}\int_{\mathbb{R}}\varphi(F_{Y^{\prime}|X^{\prime}=x_{1}}(y)-F_{Y^{\prime}|X^{\prime}=x_{2}}(y))\mathrm{\,d}\mathbb{P}^{Y^{\prime}}(y)\mathrm{\,d}(\mathbb{P}^{Y^{\prime}}\otimes\mathbb{P}^{Y^{\prime}})(y_{1},y_{2}):(X^{\prime},Y^{\prime})\in\mathcal{R}(Y)\right\\}\,,$
(21)

whenever the supremum exists and is positive.

For deriving sufficient conditions on $\varphi$ in order for
$\Lambda_{\varphi}$ in (20) to characterize independence and perfect directed
dependence, the following simple lemma is key.

######  Lemma 3.1.

Let $U$ be a real-valued random variable on the probability space
$(\Omega,\mathcal{A},\mathbb{P})$ and let $\mathcal{B}$ and $\mathcal{C}$ be
$\sigma$-algebras with
$\mathcal{B}\subseteq\mathcal{C}\subseteq\mathcal{A}\,.$ Then
$\mathbb{E}[U|\mathcal{B}]\leq_{cx}\mathbb{E}[U|\mathcal{C}]\,,$ where
$\leq_{cx}$ denotes the convex order.

###### Proof.

Let $\varphi$ be a convex function. Then, whenever the following expectations
exist, one has

$\displaystyle\mathbb{E}[\varphi(\mathbb{E}[U|\mathcal{B}])]=\mathbb{E}[\varphi(\mathbb{E}[\mathbb{E}[U|\mathcal{C}]|\mathcal{B}])]\leq\mathbb{E}[\mathbb{E}[\varphi(\mathbb{E}[U|\mathcal{C}])|\mathcal{B}]]=\mathbb{E}[\varphi(\mathbb{E}[U|\mathcal{C}])]$
(22)

applying the tower property and Jensen’s inequality for conditional
expectation. ∎

Having this we can now formulate the following main result of this section.

######  Theorem 3.2 (Measures of directed dependence).

Assume that $\varphi:[-1,1]\rightarrow\mathbb{R}$ is convex and strictly
convex in $0$ with $\varphi(0)=0$ and that $Y$ is non-degenerate. Then the
normalizing constant $\alpha_{\varphi}$ in (21) is given by

$\displaystyle\alpha_{\varphi}=\int_{\mathbb{R}^{2}}\int_{\mathbb{R}}\varphi\left(\mathds{1}_{\\{y_{1}\leq
y\\}}-\mathds{1}_{\\{y_{2}\leq
y\\}}\right)\mathrm{\,d}\mathbb{P}^{Y}(y)\mathrm{\,d}(\mathbb{P}^{Y}\otimes\mathbb{P}^{Y})(y_{1},y_{2})$
(23)

and $\Lambda_{\varphi}(Y|X)$ defined according to (20) is a dependence
measure, i.e.,

  1. (i)

$\Lambda_{\varphi}(Y|X)\in[0,1]\,,$

  2. (ii)

$\Lambda_{\varphi}(Y|X)=0$ if, and only if $X$ and $Y$ are independent,

  3. (iii)

$\Lambda_{\varphi}(Y|X)=1$ if, and only if $Y$ is perfectly dependent on $X$.

###### Proof.

(i): For every $y\in\mathbb{R}$ and every $c\in[0,1]\,,$ applying Lemma 3.1
directly yields

$\displaystyle\int_{\Omega}\varphi\left(\mathbb{E}[\mathds{1}_{\\{Y\leq
y\\}}|X]-c\right)\mathrm{\,d}\mathbb{P}\leq\int_{\Omega}\varphi\left(\mathbb{E}[\mathds{1}_{\\{Y\leq
y\\}}|(X,Y)]-c\right)\mathrm{\,d}\mathbb{P}=\int_{\Omega}\varphi\left(\mathds{1}_{\\{Y\leq
y\\}}-c\right)\mathrm{\,d}\mathbb{P}\,.$

Hence, using disintegration, Jensen’s inequality and Fubini’s theorem it
follows that

$\displaystyle 0$ $\displaystyle=\varphi(0)=\int_{\mathbb{R}}\mathbb{P}(Y\leq
y)-\mathbb{P}(Y\leq y)\,\mathrm{\,d}\mathbb{P}^{Y}(y)$
$\displaystyle=\varphi\left(\int_{\mathbb{R}}\int_{\mathbb{R}^{2}}(F_{Y|X=x_{1}}(y)-F_{Y|X=x_{2}}(y))\mathrm{\,d}(P^{X}\otimes\mathbb{P}^{X})(x_{1},x_{2})\mathrm{\,d}\mathbb{P}^{Y}(y)\right)$
$\displaystyle\leq\int_{\mathbb{R}^{2}}\int_{\mathbb{R}}\varphi\left(F_{Y|X=x_{1}}(y)-F_{Y|X=x_{2}}(y)\right)\mathrm{\,d}\mathbb{P}^{X}(x_{1})\mathrm{\,d}(\mathbb{P}^{X}\otimes\mathbb{P}^{Y})(x_{2},y)$
$\displaystyle\leq\int_{\mathbb{R}^{2}}\int_{\mathbb{R}}\varphi\left(\mathds{1}_{\\{y_{1}\leq
y\\}}-F_{Y|X=x_{2}}(y)\right)\mathrm{\,d}\mathbb{P}^{Y}(y_{1})\mathrm{\,d}(\mathbb{P}^{X}\otimes\mathbb{P}^{Y})(x_{2},y)$
$\displaystyle=\int_{\mathbb{R}^{2}}\int_{\mathbb{R}}\varphi\left(\mathds{1}_{\\{y_{1}\leq
y\\}}-F_{Y|X=x_{2}}(y)\right)\mathrm{\,d}\mathbb{P}^{X}(x_{2})\mathrm{\,d}(\mathbb{P}^{Y}\otimes\mathbb{P}^{Y})(y_{1},y)$
$\displaystyle\leq\int_{\mathbb{R}^{2}}\int_{\mathbb{R}}\varphi\left(\mathds{1}_{\\{y_{1}\leq
y\\}}-\mathds{1}_{\\{y_{2}\leq
y\\}}\right)\mathrm{\,d}\mathbb{P}^{Y}(y_{2})\mathrm{\,d}(\mathbb{P}^{Y}\otimes\mathbb{P}^{Y})(y_{1},y)$
$\displaystyle=\int_{\mathbb{R}^{2}}\int_{\mathbb{R}}\varphi\left(\mathds{1}_{\\{y_{1}\leq
y\\}}-\mathds{1}_{\\{y_{2}\leq
y\\}}\right)\mathrm{\,d}\mathbb{P}^{Y}(y)\mathrm{\,d}(\mathbb{P}^{Y}\otimes\mathbb{P}^{Y})(y_{1},y_{2})$
$\displaystyle=\sup\left\\{\int_{\mathbb{R}^{2}}\int_{\mathbb{R}}\varphi(F_{Y^{\prime}|X^{\prime}=x_{1}}(y)-F_{Y^{\prime}|X^{\prime}=x_{2}}(y))\mathrm{\,d}\mathbb{P}^{Y^{\prime}}(y)\mathrm{\,d}(\mathbb{P}^{Y^{\prime}}\otimes\mathbb{P}^{Y^{\prime}})(y_{1},y_{2}):(X^{\prime},Y^{\prime})\in\mathcal{R}(Y)\right\\}$
$\displaystyle=\alpha_{\varphi}\,,$

where the supremum is attained whenever $Y=f(X)$ almost surely for some
measurable $f$. This implies (23) and yields
$\Lambda_{\varphi}(Y|X)\in[0,1]\,.$
(ii): $X$ and $Y$ are independent, if and only if
$F_{Y|X=x}(\cdot)=F_{Y}(\cdot)$ for $\mathbb{P}^{X}$-almost all $x$. The
latter, however, is equivalent to the integrand in (20) being zero
$\mathbb{P}^{X}\otimes\mathbb{P}^{X}$-almost everywhere since $\varphi$ is
strictly convex in $0$ and $\varphi(0)=0\,$ holds.
(iii) Suppose that $Y=f(X)$ almost surely for some measurable
$f:\mathbb{R}\rightarrow\mathbb{R}$. Then obviously
$F_{Y|X=x}(y)=\mathds{1}_{\\{f(x)\leq y\\}}$, so applying Fubini’s theorem and
change of coordinates yields

$\displaystyle\alpha_{\varphi}\,\Lambda_{\varphi}(Y|X)$
$\displaystyle=\int_{\mathbb{R}}\int_{\mathbb{R}^{2}}\varphi(F_{Y|X=x_{1}}(y)-F_{Y|X=x_{2}}(y))\mathrm{\,d}(\mathbb{P}^{X}\otimes\mathbb{P}^{X})(x_{1},x_{2})\mathrm{\,d}\mathbb{P}^{Y}(y)$
(24)
$\displaystyle=\int_{\mathbb{R}}\int_{\mathbb{R}^{2}}\varphi(\mathds{1}_{\\{f(x_{1})\leq
y\\}}-\mathds{1}_{\\{f(x_{2})\leq
y\\}})\mathrm{\,d}(\mathbb{P}^{X}\otimes\mathbb{P}^{X})(x_{1},x_{2})\mathrm{\,d}\mathbb{P}^{Y}(y)$
$\displaystyle=\int_{\mathbb{R}}\int_{\mathbb{R}^{2}}\varphi(\mathds{1}_{\\{y_{1}\leq
y\\}}-\mathds{1}_{\\{y_{2}\leq y\\}})\mathrm{\,d}(\mathbb{P}^{f\circ
X}\otimes\mathbb{P}^{f\circ X})(y_{1},y_{2})\mathrm{\,d}\mathbb{P}^{Y}(y)$
$\displaystyle=\alpha_{\varphi}.$

Since $Y$ is non-degenerate, (ii) and (23) imply that $\alpha_{\varphi}$ is
positive and finite, so $\Lambda_{\varphi}(Y|X)=1$ holds.
It remains to prove the reverse implication, i.e., that
$\Lambda_{\varphi}(Y|X)=1$ implies $Y=f(X)$ almost surely for some $f$, which
can be done as follows: First, we show that the condition
$\Lambda_{\varphi}(Y|X)=1$ implies the existence of some $\mathbb{P}^{Y}$-null
set $N$ such that for fixed $y\in N^{c}$ there exists a $\mathbb{P}^{X}$-null
set $N_{y}$ with

$\displaystyle F_{Y|X=x}(y)\in\\{0,1\\}\quad\text{for all }x\in N_{y}^{c}\,.$
(25)

We then prove that there exists some $\mathbb{P}^{X}$-null set $N_{*}$
fulfilling

$\displaystyle F_{Y|X=x}(y)\in\\{0,1\\}\quad\text{for all
}y\in\mathbb{R}\text{ and for all }x\in N_{*}^{c}\,.$ (26)

Having this, proceeding like in the proofs of [4, Theorem 9.2. (iii)] or [37,
Lemma 12] yields complete dependence of $Y$ on $X$, which completes the proof.
In order to show (25), fix $y\in\mathbb{R}$ and consider
$a:\mathbb{R}\rightarrow\mathbb{R}$, defined by $a(x)=\mathbb{P}(Y\leq y\mid
X=x)$. We first establish an upper bound for the mapping
$(x_{1},x_{2})\mapsto\varphi\big{(}a(x_{1})-a(x_{2})\big{)}$ and procced as
follows: Setting

$\displaystyle c_{1}$
$\displaystyle:=a(x_{1})-\min\\{a(x_{1}),a(x_{2})\\}\in[0,1]$ $\displaystyle
c_{-1}$ $\displaystyle:=a(x_{2})-\min\\{a(x_{1}),a(x_{2})\\}\in[0,1]$
$\displaystyle c_{0}$
$\displaystyle:=1-a(x_{1})-a(x_{2})+2\min\\{a(x_{1}),a(x_{2})\\}\in[0,1]$

we obviously have $c_{1}+c_{-1}+c_{0}=1$, so using convexity yields

$\displaystyle\varphi(a(x_{1})-a(x_{2}))$ (27)
$\displaystyle=\varphi\big{(}1\cdot(a(x_{1})-\min\\{a(x_{1}),a(x_{2})\\})-1\cdot(a(x_{2})-\min\\{a(x_{1}),a(x_{2})\\})+0\cdot
c_{0}\big{)}$
$\displaystyle\leq\varphi(1)(a(x_{1})-\min\\{a(x_{1}),a(x_{2})\\})+\varphi(-1)(a(x_{2})-\min\\{a(x_{1}),a(x_{2})\\})+\underbrace{\varphi(0)}_{=0}\cdot
c_{0}$
$\displaystyle=\varphi(1)\,a(x_{1})+\varphi(-1)\,a(x_{2})-\underbrace{(\varphi(1)+\varphi(-1))}_{>0}\,\underbrace{\min\\{a(x_{1}),a(x_{2})\\}}_{\geq
a(x_{1})a(x_{2})}$
$\displaystyle\leq\varphi(1)\,a(x_{1})+\varphi(-1)\,a(x_{2})-(\varphi(1)+\varphi(-1))\,a(x_{1})a(x_{2})$
$\displaystyle=\varphi(1)\,a(x_{1})(1-a(x_{2}))+\varphi(-1)\,a(x_{2})(1-a(x_{1}))$
$\displaystyle=\int_{\mathbb{R}^{2}}\varphi\left(\mathds{1}_{\\{y_{1}\leq
y\\}}-\mathds{1}_{\\{y_{2}\leq
y\\}}\right)\mathrm{\,d}(\mathbb{P}^{Y|X=x_{1}}\otimes\mathbb{P}^{Y|X=x_{2}})(y_{1},y_{2}).$

Integrating both sides over $\mathbb{R}^{2}$ with respect to
$\mathbb{P}^{X}\otimes\mathbb{P}^{X}$ and using disintegration it follows that

$\displaystyle\int_{\mathbb{R}^{2}}\varphi(a(x_{1})-a(x_{2}))\mathrm{\,d}\mathbb{P}^{X}\otimes\mathbb{P}^{X}(x_{1},x_{2})$
(28)
$\displaystyle\leq\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\varphi\left(\mathds{1}_{\\{y_{1}\leq
y\\}}-\mathds{1}_{\\{y_{2}\leq
y\\}}\right)\mathrm{\,d}(\mathbb{P}^{Y|X=x_{1}}\otimes\mathbb{P}^{Y|X=x_{2}})(y_{1},y_{2})\,\mathrm{\,d}\mathbb{P}^{X}\otimes\mathbb{P}^{X}(x_{1},x_{2})$
$\displaystyle=\int_{\mathbb{R}}\int_{\mathbb{R}}\int_{\mathbb{R}}\int_{\mathbb{R}}\varphi\left(\mathds{1}_{\\{y_{1}\leq
y\\}}-\mathds{1}_{\\{y_{2}\leq
y\\}}\right)\mathrm{\,d}\mathbb{P}^{Y|X=x_{1}}(y_{1})\mathrm{\,d}\mathbb{P}^{X}(x_{1})\mathrm{\,d}\mathbb{P}^{Y|X=x_{2}}(y_{2})\mathrm{\,d}\mathbb{P}^{X}(x_{2})$
$\displaystyle=\int_{\mathbb{R}^{2}}\varphi(\mathds{1}_{\\{y_{1}\leq
y\\}}-\mathds{1}_{\\{y_{2}\leq
y\\}})\mathrm{\,d}\mathbb{P}^{Y}\otimes\mathbb{P}^{Y}(y_{1},y_{2}).$

Consequently, if equality holds in (28), then for
$\mathbb{P}^{X}\otimes\mathbb{P}^{X}$-almost all
$(x_{1},x_{2})\in\mathbb{R}^{2}$ inequality (27) becomes an equality. In
particular, there exists some Borel set $G\subseteq\mathbb{R}^{2}$ with
$\mathbb{P}^{X}\otimes\mathbb{P}^{X}(G)=1$ such that

$\displaystyle\min\\{a(x_{1}),a(x_{2})\\}=a(x_{1})a(x_{2})$

holds for all $(x_{1},x_{2})\in G$. Considering that the latter is equivalent
to the fact that $a(x_{1})\in\\{0,1\\}$ or $a(x_{2})\in\\{0,1\\}$ for all
$(x_{1},x_{2})\in G$ we conclude that $a(x)\in\\{0,1\\}$ for
$\mathbb{P}^{X}$-almost every $x\in\mathbb{R}$. Summing up, for every fixed
$y\in\mathbb{R}$ we have shown that equality in (28) implies that
$a(x)=\mathbb{P}(Y\leq y\mid X=x)=F_{Y|X=x}(y)\in\\{0,1\\}$ for
$\mathbb{P}^{X}$-almost every $x\in\mathbb{R}$.
Now, to prove (25), suppose that $\Lambda_{\varphi}(Y|X)=1$, i.e., that we
have

$\displaystyle\begin{split}&\int_{\mathbb{R}}\int_{\mathbb{R}^{2}}\varphi(a(x_{1})-a(x_{2}))\mathrm{\,d}\mathbb{P}^{X}\otimes\mathbb{P}^{X}(x_{1},x_{2})\mathrm{\,d}\mathbb{P}^{Y}(y)\\\
&=\int_{\mathbb{R}}\int_{\mathbb{R}^{2}}\varphi(\mathds{1}_{\\{y_{1}\leq
y\\}}-\mathds{1}_{\\{y_{2}\leq
y\\}}\mathrm{\,d}\mathbb{P}^{Y}\otimes\mathbb{P}^{Y}(y_{1},y_{2})\mathrm{\,d}\mathbb{P}^{Y}(y)\,.\end{split}$
(29)

Then, due to (28), there exists some Borel set $E\subseteq\mathbb{R}$ with
$\mathbb{P}^{Y}(E)=1$ such that for every $y\in E$ we have equality in (28).
Due to the previous step, for $N:=E^{c}$ (25) holds true for all $y\in E\,,$
i.e., for every $y\in E$ there exists a $\mathbb{P}^{X}$-null set $N_{y}$ with
$F_{Y|X=x}(y)\in\\{0,1\\}$ for all $x\in N_{y}^{c}$.
In order to prove (26) first note that by disintegration we also have
$\mathbb{P}^{Y|X=x}(E)=1$ for $\mathbb{P}^{X}$-almost every
$x\in\mathbb{R}\,.$ Hence we may proceed as follows: Define the set
$N\in\mathcal{B}(\mathbb{R}^{2})$ by

$\displaystyle N:=\\{(x,y)\in\mathbb{R}^{2}:F_{Y|X=x}(y)\in(0,1)\\}$

and suppose that $\mathbb{P}^{(X,Y)}(N)>0$. Setting
$N_{x}:=\\{y\in\mathbb{R}:(x,y)\in N\\}$ disintegration yields

$\displaystyle
0<\mathbb{P}^{(X,Y)}(N)=\int_{\mathbb{R}}\mathbb{P}^{Y|X=x}(N_{x})\mathrm{\,d}\mathbb{P}^{X}(x),$

so the set $O\subseteq\mathcal{\mathbb{R}}$, given by
$O=\\{x\in\mathbb{R}:\mathbb{P}^{Y|X=x}(N_{x})>0\textrm{ and
}\mathbb{P}^{Y|X=x}(E)=1\\}$ fulfills $\mathbb{P}^{X}(O)>0$. For every $x\in
O$ we therefore have $N_{x}\cap E\neq\emptyset$, which is absurd.
Consequently, $\mathbb{P}^{(X,Y)}(N)=0$, whence $\mathbb{P}^{(X,Y)}(N^{c})=1$
holds. Disintegrating once more yields a $\mathbb{P}^{X}$-null set $N_{*}$
such that for all $x\in N_{*}^{c}$ the conditional distribution function
$y\mapsto F_{Y|X=x}(y)$ only attains values in $\\{0,1\\}$, which proves (26).
∎

######  Remark 3.3.

  1. (a)

If $\varphi(x)=x^{2}$, then $\Lambda_{\varphi}$ in (20) reduces to
Chatterjee’s coefficient of correlation, see Example 1. However, in general,
the comparison of conditional distributions via $\Lambda_{\varphi}$ does not
boil down to comparing conditional and unconditional distribution, see Example
4. In other words, Theorem 3.2 provides a new class of dependence measures.

  2. (b)

In Theorem 3.2, the assumption of strict convexity of $\varphi$ in $0$ is on
the one hand necessary for the characterization of independence. On the other
hand, it is also used in the proof that $\Lambda_{\varphi}(Y|X)=1$ implies
complete dependence. Note that Theorem 3.2 only requires that $\varphi$ is
strictly convex in $0$ but not necessarily strictly convex on the full
interval $[-1,1]$. Hence, each of the functions $x\mapsto\max\\{x,0\\}\,,$
$x\mapsto\exp(cx)-1$ for $c\neq 0\,,$ and $x\mapsto|x|^{p}\,,$ $p\geq 1\,,$
satisfy the assumptions of Theorem 3.2. Also note that as it is the case for
$\psi$ in Theorem 2.2 also $\varphi$ may attain negative values.

  3. (c)

The second part of the proof of Theorem 3.2(iii) can also be established along
the lines of the proof of the second part of Theorem 2.2(iii). However, we
opted for a more constructive proof which provides additional insight on
properties of $\Lambda_{\varphi}$. Instead of the denseness argument here we
use the fact that for independent $X_{1}$ and $X_{2}$ with
$X_{1}\stackrel{{\scriptstyle\mathrm{d}}}{{=}}X_{2}\stackrel{{\scriptstyle\mathrm{d}}}{{=}}X$,
the random variable $F_{Y|X=X_{1}}(y)-F_{Y|X=X_{2}}(y)$ obtained from (24) is
bounded on $[-1,1]$ for every $y\in\mathbb{R}$ and the support of the extremal
elements w.r.t. convex order is $\mathbb{P}^{Y}$-almost surely the set
$\\{-1,0,1\\}\,.$

###  3.2 Additional properties of $\Lambda_{\varphi}$

Convexity of the function $\varphi$ yields the so-called _data processing
inequality_ which states that applying a transformation to the explanatory
variable $X$ can not increase the extent of dependence, see [7] for a detailed
discussion. Note that strict convexity in $0$ is not required for the
following results.

######  Proposition 3.4 (Data processing inequality).

Assume that $\varphi$ is convex on $[-1,1]$. Then $\Lambda_{\varphi}$ defined
according to eq. (20) fulfills the data processing inequality, i.e.,

$\displaystyle\Lambda_{\varphi}(Y|Z)\leq\Lambda_{\varphi}(Y|X)$

for all $X,Y,Z$ such that $Y$ and $Z$ are conditionally independent given $X$.
In particular,

$\displaystyle\Lambda_{\varphi}(Y|h(X))\leq\Lambda_{\varphi}(Y|X)$

for all measurable functions $h:\mathbb{R}\rightarrow\mathbb{R}$.

###### Proof.

Assume that $Y$ and $Z$ are conditionally independent given $X$. Proceeding as
in the proof of the second assertion of Lemma 3.5 in [18] and using
disintegration it follows that for every fixed $y\in\mathbb{R}$ we have

$\displaystyle
F_{Y|Z=z}(y)=\int_{\mathbb{R}}F_{Y|Z=z,X=x}(y)\mathrm{\,d}\mathbb{P}^{X|Z=z}(x)=\int_{\mathbb{R}}F_{Y|X=x}(y)\mathrm{\,d}\mathbb{P}^{X|Z=z}(x)$

for $\mathbb{P}^{Z}$-almost all $z\in\mathbb{R}$. Hence, using convexity of
$\varphi$, Jensen’s inequality, and disintegration once more yields

$\displaystyle\int_{\mathbb{R}^{2}}\int_{\mathbb{R}}\varphi(F_{Y|Z=z_{1}}(y)-F_{Y|Z=z_{2}}(y))\mathrm{\,d}\mathbb{P}^{Y}(y)\mathrm{\,d}(\mathbb{P}^{Z}\otimes\mathbb{P}^{Z})(z_{1},z_{2})$
$\displaystyle=\int_{\mathbb{R}^{2}}\int_{\mathbb{R}}\varphi\left(\int_{\mathbb{R}}F_{Y|X=x_{1}}(y)\mathrm{\,d}\mathbb{P}^{X|Z=z_{1}}(x_{1})-\int_{\mathbb{R}}F_{Y|X=x_{2}}(y)\mathrm{\,d}\mathbb{P}^{X|Z=z_{2}}(x_{2})\right)\mathrm{\,d}\mathbb{P}^{Y}(y)\mathrm{\,d}(\mathbb{P}^{Z}\otimes\mathbb{P}^{Z})(z_{1},z_{2})$
$\displaystyle=\int_{\mathbb{R}^{2}}\int_{\mathbb{R}}\varphi\left(\int_{\mathbb{R}^{2}}F_{Y|X=x_{1}}(y)-F_{Y|X=x_{2}}(y)\mathrm{\,d}(\mathbb{P}^{X|Z=z_{1}}\otimes\mathbb{P}^{X|Z=z_{2}})(x_{1},x_{2})\right)\mathrm{\,d}\mathbb{P}^{Y}(y)\mathrm{\,d}(\mathbb{P}^{Z}\otimes\mathbb{P}^{Z})(z_{1},z_{2})$
$\displaystyle\leq\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\int_{\mathbb{R}}\varphi(F_{Y|X=x_{1}}(y)-F_{Y|X=x_{2}}(y))\mathrm{\,d}\mathbb{P}^{Y}(y)\mathrm{\,d}(\mathbb{P}^{X|Z=z_{1}}\otimes\mathbb{P}^{X|Z=z_{2}})(x_{1},x_{2})\mathrm{\,d}(\mathbb{P}^{Z}\otimes\mathbb{P}^{Z})(z_{1},z_{2})$
$\displaystyle=\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\int_{\mathbb{R}}\varphi(F_{Y|X=x_{1}}(y)-F_{Y|X=x_{2}}(y))\mathrm{\,d}\mathbb{P}^{Y}(y)\mathrm{\,d}\mathbb{P}^{(X,Z)}(x_{1},z_{1})\mathrm{\,d}\mathbb{P}^{(X,Z)}(x_{2},z_{2})$
$\displaystyle=\int_{\mathbb{R}^{2}}\int_{\mathbb{R}}\varphi(F_{Y|X=x_{1}}(y)-F_{Y|X=x_{2}}(y))\mathrm{\,d}\mathbb{P}^{Y}(y)\mathrm{\,d}(\mathbb{P}^{X}\otimes\mathbb{P}^{X})(x_{1},x_{2})\,,$

which completes the proof. ∎

As a consequence of the data processing inequality, $\Lambda_{\varphi}$
satisfies the property of _self-equitability_ introduced in [24] (see also
[10]) which roughly states that for $(X,Y)$ in a regression setting
$Y=f(X)+\varepsilon\,,$ the function $\Lambda_{\varphi}$ should only depend on
the distribution of the noise $\varepsilon$ and not on the specific form of
the function $f$, see [35].

######  Corollary 3.5 (Self-equitability).

Assume that $\varphi$ is convex on $[-1,1]\,.$ Then $\Lambda_{\varphi}$
defined according to (20) is _self-equitable_ , i.e.,

$\displaystyle\Lambda_{\varphi}(Y|h(X))=\Lambda_{\varphi}(Y|X)$

for all $(X,Y)$ and all measurable functions $h$ such that $Y$ and $X$ are
conditionally independent given $h(X)$.

The following result provides additional useful invariance properties for
$\Lambda_{\varphi}$. Note that for these properties to hold we do not even
need $\varphi$ to be convex.

######  Proposition 3.6 (Invariances).

$\Lambda_{\varphi}$ satisfies the invariance properties (P3), (P4), and (P5),
i.e.,

  1. (i)

$\Lambda_{\varphi}(Y|X)=\Lambda_{\varphi}(Y|h(X))$ for all bijective functions
$h:\mathbb{R}\rightarrow\mathbb{R}$,

  2. (ii)

$\Lambda_{\varphi}(Y|X)=\Lambda_{\varphi}(g(Y)|X)$ for all strictly increasing
functions $g:\mathbb{R}\rightarrow\mathbb{R}$,

  3. (iii)

$\Lambda_{\varphi}(Y|X)=\Lambda_{\varphi}(Y^{\prime}|X^{\prime})$ whenever
$(X,Y)\stackrel{{\scriptstyle\mathrm{d}}}{{=}}(X^{\prime},Y^{\prime})$,

###### Proof.

(i): Let $(X^{\prime},Y^{\prime})$ be an independent copy of $(X,Y)$. Since
$h$ is bijective, the $\sigma$-algebras generated by $X$ and $h(X)$ coincide.
Using change of coordinates it therefore follows that

$\displaystyle\Lambda_{\varphi}(Y|X)$
$\displaystyle=\int_{\mathbb{R}}\int_{\Omega}\varphi\left(\mathbb{E}[\mathds{1}_{\\{Y\leq
y\\}}|X](\omega)-\mathbb{E}[\mathds{1}_{\\{Y^{\prime}\leq
y\\}}|X^{\prime}](\omega)\right)\mathrm{\,d}\mathbb{P}(\omega)\mathrm{\,d}\mathbb{P}^{Y}(y)$
$\displaystyle=\int_{\mathbb{R}}\int_{\Omega}\varphi\left(\mathbb{E}[\mathds{1}_{\\{Y\leq
y\\}}|h(X)](\omega)-\mathbb{E}[\mathds{1}_{\\{Y^{\prime}\leq
y\\}}|h(X^{\prime})](\omega)\right)\mathrm{\,d}\mathbb{P}(\omega)\mathrm{\,d}\mathbb{P}^{Y}(y)=\Lambda_{\varphi}(Y|h(X))\,.$

(ii): Since there exists some measurable function
$k\colon\mathbb{R}^{2}\to[0,1]$ such that $F_{Y|X=x}(y)=k(x,F_{Y}(y))$ for
$\mathbb{P}^{X}$-almost all $x\in\mathbb{R}$ and for all $y\in\mathbb{R}\,,$
see [3, Theorem 2.2], using change of coordinates and that $g$ is strictly
increasing it follows that ($\lambda$ denoting the Lebesgue measure, $F^{-}$
the quasi-inverse of a distribution function $F$).

$\displaystyle\Lambda_{\varphi}(Y|X)$
$\displaystyle=\int_{\mathbb{R}^{2}}\int_{(0,1)}\varphi\left(k\left(x_{1},F_{Y}\circ
F_{Y}^{-}(u)\right)-k(x_{2},F_{Y}\circ
F_{Y}^{-}(u))\right)\mathrm{\,d}\lambda(u)\mathrm{\,d}\mathbb{P}^{X}\otimes\mathbb{P}^{X}(x_{1},x_{2})$
$\displaystyle=\int_{\mathbb{R}^{2}}\int_{(0,1)}\varphi\left(k\left(x_{1},F_{g(Y)}\circ
F_{g(Y)}^{-}(u)\right)-k\left(x_{2},F_{g(Y)}\circ
F_{g(Y)}^{-}(u)\right)\right)\mathrm{\,d}\lambda(u)\mathrm{\,d}\mathbb{P}^{X}\otimes\mathbb{P}^{X}(x_{1},x_{2})$
$\displaystyle=\Lambda_{\varphi}(g(Y)|X)).$

Statement (iii) is trivial. ∎

Considering that $\Lambda(Y|X)$ according to Theorem 3.6(i) is invariant with
respect to bijective transformations of $X$, in order to establish some
general ordering results for $\Lambda_{\varphi}$ we need to work with
rearrangement invariant orders. To this end, for integrable functions
$f,g\colon(0,1)\to\mathbb{R}$ with
$\int_{0}^{1}f(t)\mathrm{\,d}\lambda(t)=\int_{0}^{1}g(t)\mathrm{\,d}\lambda(t)$
consider the Schur order $\prec_{S}$, defined by

$\displaystyle
f\prec_{S}g\quad:\,\Longleftrightarrow\quad\int_{0}^{x}f^{*}(t)\mathrm{\,d}t\leq\int_{0}^{x}g^{*}(t)\mathrm{\,d}t\quad\text{for
all }x\in(0,1)\,,$ (30)

where $h^{*}$ denotes decreasing rearrangement of a function
$h\colon(0,1)\to\mathbb{R}\,,$ i.e., the $\lambda$-almost everywhere unique
decreasing function $h^{*}$ such that $\lambda(\\{h^{*}\leq
t\\})=\lambda(\\{h\leq t\\})$ for all $t\in\mathbb{R}\,.$ Denote by
$q_{Z}(u):=\inf\\{z\in\mathbb{R}:\mid F_{Z}(z)\geq u\\}=F_{Z}^{-}(u)$ the
quasi-inverse of the distribution function $F_{Z}$ of $Z$. Then, for $Y$ and
$Y^{\prime}$ having continuous distribution functions, the Schur order
$(Y|X)\leq_{S}(Y^{\prime}|X^{\prime})$ for conditional distributions is
defined by

$\displaystyle\mathbb{E}\left[\mathds{1}_{\\{Y\leq q_{Y}(v)\\}}\mid
X=q_{X}(\cdot)\right]\leq_{S}\mathbb{E}\left[\mathds{1}_{\\{Y^{\prime}\leq
q_{Y^{\prime}}(v)\\}}\mid
X^{\prime}=q_{X^{\prime}}(\cdot)\right]~{}~{}~{}\text{for all }v\in(0,1)\,,$
(31)

see [2] as well as [1, 3, 33, 35]. Hence, the Schur-order for conditional
distributions compares the variability of conditional distribution functions
in the conditioning variable.

For bivariate random vectors $(U,V)$ and $(U^{\prime},V^{\prime})\,,$ denote
by $(U,V)\leq_{lo}(U^{\prime},V^{\prime})$ the lower orthant order, i.e., the
pointwise comparison of the associated distribution functions. Further, the
bivariate random vector $(U,V)$ is said to be stochastically increasing (SI)
if the conditional distribution $V|U=u$ is stochastically increasing in $u\,,$
i.e., $\mathbb{E}[f(V)|U=u]$ is increasing in $u$ for all increasing functions
$f$ such that the expectations exist. If $(U^{\prime},V^{\prime})$ is SI and
if $U\stackrel{{\scriptstyle\mathrm{d}}}{{=}}U^{\prime}$ and
$V\stackrel{{\scriptstyle\mathrm{d}}}{{=}}V^{\prime}\,,$ then
$(V|U)\leq_{S}(V^{\prime}|U^{\prime})$ implies
$(U,V)\leq_{lo}(U^{\prime},V^{\prime})\,.$ If additionally $(U,V)$ is SI, then
$(V|U)\leq_{S}(V^{\prime}|U^{\prime})$ and
$(U,V)\leq_{lo}(U^{\prime},V^{\prime})$ are equivalent, see [3, Lemma 3.16].

The following result provides simple conditions for monotonicity of
$\Lambda_{\varphi}\,.$

######  Proposition 3.7 (Monotonicity).

Suppose that $(Y|X)\leq_{S}(Y^{\prime}|X^{\prime})$ and that $Y,Y^{\prime}$
have continuous distribution function. Then
$\Lambda_{\varphi}(Y|X)\leq\Lambda_{\varphi}(Y^{\prime}|X^{\prime})$ for all
convex functions $\varphi$ such that the integrals exist.

###### Proof.

By the Hardy-Littlewood-Polya theorem (see, e.g., [32, Theorem 3.21] we have
that $(Y|X)\leq_{S}(Y^{\prime}|X^{\prime})$ is equivalent to

$\displaystyle\int_{0}^{1}\phi(F_{Y|X=q_{X}(u)}(q_{Y}(v)))\mathrm{\,d}\lambda(u)\leq\int_{0}^{1}\phi(F_{Y^{\prime}|X^{\prime}=q_{X^{\prime}}(u)}(q_{Y^{\prime}}(v)))\mathrm{\,d}\lambda(u)\quad$

for all $v$ and for all convex $\phi$ such that the integrals exist. Since the
functions $z\mapsto\varphi(z-c)$ and $z\mapsto\varphi(c-z)$ are convex
whenever $\varphi$ is convex, using Fubini’s theorem and change of coordinates
repeatedly it follows that

$\displaystyle\int_{\mathbb{R}^{2}}\int_{\mathbb{R}}\varphi(F_{Y|X=x_{1}}(y)-F_{Y|X=x_{2}}(y))\mathrm{\,d}\mathbb{P}^{Y}(y)\mathrm{\,d}(\mathbb{P}^{X}\otimes\mathbb{P}^{X})(x_{1},x_{2})$
$\displaystyle=\int_{(0,1)}\int_{(0,1)}\int_{(0,1)}\varphi\left(F_{Y|X=q_{X}(u_{1})}(q_{Y}(v))-F_{Y|X=q_{X}(u_{2})}(q_{Y}(v))\right)\mathrm{\,d}\lambda(u_{1})\mathrm{\,d}\lambda(v)\mathrm{\,d}\lambda(u_{2})$
$\displaystyle\leq\int_{(0,1)}\int_{(0,1)}\int_{(0,1)}\varphi\left(F_{Y^{\prime}|X^{\prime}=q_{X^{\prime}}(u_{1})}(q_{Y^{\prime}}(v))-F_{Y^{\prime}|X^{\prime}=q_{X^{\prime}}(u_{2})}(q_{Y^{\prime}}(v))\right)\mathrm{\,d}\lambda(v)\mathrm{\,d}\lambda(u_{1})\mathrm{\,d}\lambda(u_{2})$
$\displaystyle=\int_{\mathbb{R}^{2}}\int_{\mathbb{R}}\varphi(F_{Y^{\prime}|X^{\prime}=x_{1}}(y)-F_{Y^{\prime}|X^{\prime}=x_{2}}(y))\mathrm{\,d}\mathbb{P}^{Y^{\prime}}(y)\mathrm{\,d}(\mathbb{P}^{X^{\prime}}\otimes\mathbb{P}^{X^{\prime}})(x_{1},x_{2})$

which implies
$\Lambda_{\varphi}(Y|X)\leq\Lambda_{\varphi}(Y^{\prime}|X^{\prime})\,.$ ∎

######  Remark 3.8.

  1. (a)

Minimal and maximal elements with respect to the Schur order for conditional
distributions characterize independence and complete dependence, respectively,
see [2, Theorem 3.5]. Hence, as a consequence of Theorem 3.7, if $Y$ is
completely depend on $X$ and if $\varphi$ is convex, then
$\Lambda_{\varphi}(Y|X)=1\,,$ using that $\alpha_{\varphi}\in(0,\infty)\,.$ If
additionally $\varphi$ is strictly convex in $0$ and if $\varphi(0)=0\,,$ then
Theorem 3.2 provides the reverse implication, i.e., that
$\Lambda_{\varphi}(Y|X)=1$ implies that $Y=f(X)$ almost surely.

  2. (b)

Various well-known (sub-)families of bivariate distributions are
stochastically increasing and increasing with respect to the lower orthant
order (implying that the associated conditional distributions are increasing
with respect to the Schur order for conditional distributions). Examples are
the bivariate normal distribution with respect to the non-negative correlation
parameter, several Archimedean copula families such as the Clayton, Gumbel or
Frank family, or various extreme value copula families, see, e.g., [28,
Chapter 4], [27], and [3, Examples 3.18 and 3.19].

###  3.3 Continuous setting and $L^{p}$-version

From now on we will assume that $(X,Y)$ has continuous bivariate distribution
function since continuity is necessary for having uniqueness in Sklar’s
theorem (see [28]) and both for deriving continuity results of the functional
$\Lambda_{\varphi}$ and for the the estimation procedure studied in the next
section. Furthermore, in the sequel (following [22]) we will write conditional
distributions as Markov kernels, i.e., $\mathbb{P}^{Y|X=x}(\cdot)=K(x,\cdot)$,
let $C$ denote the (unique) copula underlying $(X,Y)$ and $K_{C}(\cdot,\cdot)$
as the Markov kernel of $C$. Notice that according to [26, Lemma 1] (also see
[3, Theorem 2.2]), starting with a Markov kernel $K_{C}(\cdot,\cdot)$ and
setting

$K(x,(-\infty,y]):=K_{C}(F_{X}(x),[0,F_{Y}(y)])$

yields a (version of the) Markov kernel of $(X,Y)$. In the continuous setting,
the functional $\Lambda_{\varphi}$ only depends on the underlying copula $C$
since for every continuous $\varphi\colon[-1,1]\to\mathbb{R}$ using change of
coordinates we have

$\displaystyle\Lambda_{\varphi}(Y|X)$
$\displaystyle=\int_{\mathbb{R}^{2}}\int_{\mathbb{R}}\varphi\big{(}F_{Y|X=x_{1}}(y)-F_{Y|X=x_{2}}(y)\big{)}\mathrm{\,d}\mathbb{P}^{X}\otimes\mathbb{P}^{X}(x_{1},x_{2})\mathrm{\,d}\mathbb{P}^{Y}(y)$
$\displaystyle=\int_{\mathbb{R}^{2}}\int_{\mathbb{R}}\varphi\big{(}K_{C}(F_{X}(x_{1}),[0,F_{Y}(y)])-K_{C}(F_{X}(x_{2}),[0,F_{Y}(x)])\big{)}\mathrm{\,d}\mathbb{P}^{X}\otimes\mathbb{P}^{X}(x_{1},x_{2})\mathrm{\,d}\mathbb{P}^{Y}(y)$
$\displaystyle=\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\varphi\big{(}K_{C}(u_{1},[0,v])-K_{C}(u_{2},[0,v]\big{)}\mathrm{\,d}\lambda(v)\mathrm{\,d}\lambda(u_{1})\mathrm{\,d}\lambda(u_{2}).$
(32)

Following [23] we say that a sequence a sequence $(C_{n})_{n\in\mathbb{N}}$ of
bivariate copulas _converges weakly conditional_ to a bivariate copula $C$ and
write $C_{n}\xrightarrow{wcc}C\,,$ if there exists some Borel set
$\Gamma\subseteq[0,1]$ with $\lambda(\Gamma)=1$ such that the probability
measures $(K_{C_{n}}(u,\cdot))_{n\in\mathbb{N}}$ converges weakly to the
probability measure $K_{C}(u,\cdot)$ for every $x\in\Gamma$. The following
result provides sufficient conditions for continuity of the functional
$\Lambda_{\varphi}$ and is straightforward to prove using eq. (3.3) and
dominated convergence.

######  Proposition 3.9 (Continuity).

Let $(X,Y),(X_{1},Y_{1}),(X_{2},Y_{2}),\ldots$ be bivariate random vectors
with continuous distribution functions and corresponding copulas
$A,A_{1},A_{2},\ldots$ and let $\varphi\colon[-1,1]\to\mathbb{R}$ be
continuous. Then $A_{n}\xrightarrow{wcc}A$ implies convergence of
$(\Lambda_{\varphi}(Y_{n}|X_{n}))_{n\in\mathbb{N}}$ to
$\Lambda_{\varphi}(Y|X)$.

######  Example 3.

A particular class of dependence measures fulfilling all of the afore-
mentioned properties is obtained when considering $\varphi(x)=|x|^{p}$ for
some $p\geq 1$. In this case $\Lambda_{\varphi}$ corresponds to the
$L^{p}$-distance of conditional distribution functions. In fact, in this case
using eq. (3.3) we have

$\displaystyle\Lambda_{p}(Y|X)$
$\displaystyle:=3\int_{[0,1]^{3}}|K_{A}(u_{1},[0,z])-K_{A}(u_{2},[0,z])|^{p}\,\mathrm{\,d}\lambda^{3}(u_{1},u_{2},z)=\Lambda_{p}(A)\,,$
(33)

where the normalizing constant according is given by

$\displaystyle\alpha_{p}:=\alpha_{\varphi}=\int_{(0,1)^{3}}|\mathds{1}_{\\{u_{1}\leq
v\\}}-\mathds{1}_{\\{u_{2}\leq
v\\}}|^{p}\mathrm{\,d}\lambda^{3}(u_{1},u_{2},v).$ (34)

Since the normalizing constant does not depend on $p$ it follows that the
mapping $p\mapsto\Lambda_{p}(Y|X)$ is decreasing in $p\in[1,\infty)$.
For $p=2$ the functional $\Lambda_{p}$ compares two conditional distributions
as in (1) and coincides with Chatterjee’s correlation coefficient (see Example
1) which compares conditional and unconditional distributions as in (2). For
$p=1\,,$ however, the following examples illustrates that the comparison of
two conditional distributions as in (1) only coincides for some specific
families with the $L^{1}$-analogue of Chatterjee’s coefficient of correlation.

######  Example 4 (FGM and Fréchet copulas).

The functional $\zeta_{1}(Y|X)$ defined by

$\displaystyle\zeta_{1}(Y|X):=\frac{1}{3}\int_{0}^{1}|F_{Y|X=x}(y)-F_{Y}(y)|\mathrm{\,d}\mathbb{P}^{Y}(y)\mathrm{\,d}\mathbb{P}^{X}(x)$
(35)

is a dependence measure introduced and studied in [37].

  1. (a)

For $\alpha\in[0,1]$ the copula

$\displaystyle C_{\alpha}(u,v):=uv+\alpha\,u(1-u)v(1-v)$

is a member of the Farlie-Gumbel-Morgenstern (FGM) family. Let $(X,Y)$ be a
random vector with continuous distribution function and underlying copula
$C_{\alpha}$. Then straightforward calculations yield

$\displaystyle\Lambda_{1}(Y|X)=\frac{\alpha}{3}\qquad\textrm{ and
}\qquad\zeta_{1}(Y|X)=\frac{\alpha}{4},$

so $\Lambda_{1}(Y|X)=\tfrac{4}{3}\,\zeta_{1}(Y|X)\neq\zeta_{1}(Y|X)$.

  2. (b)

For $\alpha\in[0,1]$ the copula

$\displaystyle C_{\alpha}(u,v):=\alpha\,M(u,v)+(1-\alpha)\,\Pi(u,v)$

is a member of the Fréchet copula family. In this case for continuous $(X,Y)$
with underlying copula $C_{\alpha}$ we get

$\displaystyle\Lambda_{1}(Y|X)=\alpha=\zeta_{1}(Y|X)\,.$

##  4 Estimating the functionals

In this section, we propose a copula-based so-called checkerboard estimator as
plug-in estimator for the functionals $\Lambda_{\psi}$ in (7) and
$\Lambda_{\varphi}$ in (20) and start with recalling some basic facts.
For every bivariate copula $A$ and $N\in\mathbb{N}$ the $N$-checkerboard
approximation of $A$ (see [25, 22]) will be denoted by $\mathfrak{Cb}_{N}(A)$.
As a consequence of [25, Corollary 3.2], for every copula $A$ the sequence
$(\mathfrak{Cb}_{N}(A))_{N\in\mathbb{N}}$ converges weakly conditional to $A$.
To assure well-definedness we will refer to the bilinear interpolation of the
subcopula resulting from Sklar’s theorem by considering the empirical
distribution functions as the empirical copula, see, e.g., [16, 18].
Plugging in the empirical copula we get the following consistency result.

######  Lemma 4.1.

Let $(X_{1},Y_{1}),(X_{2},Y_{2}),\ldots$ be a random sample from $(X,Y)$ and
assume that $(X,Y)$ has continuous distribution function $H$, underlying
copula $A$, and that $E_{n}$ denotes the empirical copula (of the first $n$
observations). Then setting $N(n):=\lfloor{n}^{s}\rfloor$ for some
$s\in(0,\frac{1}{2})$ the sequence
$(\mathfrak{Cb}_{N(n)}(E_{n}))_{n\in\mathbb{N}}$ converges weakly conditional
to $A$ with probability $1$.

###### Proof.

Choose $\tilde{\Gamma}\in\mathcal{B}([0,1])$ with $\lambda(\tilde{\Gamma})=1$
in such a way that for every $x\in\tilde{\Gamma}$ we have that
$(K_{\mathfrak{Cb}_{N}(A)}(x,\cdot))_{N\in\mathbb{N}}$ converges weakly to
$K_{A}(x,\cdot)$. Setting
$\Gamma:=\tilde{\Gamma}\setminus\bigcup_{N=2}^{\infty}\\{0,\frac{1}{N},\frac{2}{N},\ldots,\frac{N-1}{N},1\\}$
obviously yields $\lambda(\Gamma)=1$.
Fix $x\in\Gamma$ and consider some $y\in(0,1)$ with

$\lim_{N\rightarrow\infty}K_{\mathfrak{Cb}_{N}(A)}(x,[0,y])=K_{A}(x,[0,y]).$
(36)

Plugging in the empirical copula, setting
$I_{n}:=|K_{\mathfrak{Cb}_{N(n)}(E_{n})}(x,[0,y])-K_{A}(x,[0,y])|$, and using
the triangle inequality yields

$\displaystyle I_{n}$ $\displaystyle\leq$
$\displaystyle|K_{\mathfrak{Cb}_{N(n)}(E_{n})}(x,[0,y])-K_{\mathfrak{Cb}_{N(n)}(A)}(x,[0,y])|+|K_{\mathfrak{Cb}_{N(n)}(A)}(x,[0,y])-K_{A}(x,[0,y])|$

The second summand converges to $0$ for $n\rightarrow\infty$ due to (36). In
order to show that the first summand converges to $0$ almost surely, denote by
$i_{N}(x)\,,$ $N\geq 2\,,$ the unique integer in $\\{1,\ldots,N\\}$ fulfilling
$x\in(\frac{i_{N}(x)-1}{N},\frac{i_{N}(x)}{N})$.
Since $\mathfrak{Cb}_{N(n)}(E_{n})$ and $\mathfrak{Cb}_{N(n)}(A)$ are
checkerboard copulas, we have

$\displaystyle K_{\mathfrak{Cb}_{N(n)}(E_{n})}(x,[0,y])$ $\displaystyle=$
$\displaystyle
N(n)\left(E_{n}\left(\frac{i_{N(n)}(x)}{N(n)},y\right)-E_{n}\left(\frac{i_{N(n)}(x)-1}{N(n)},y\right)\right)$
$\displaystyle K_{\mathfrak{Cb}_{N(n)}(A)}(x,[0,y])$ $\displaystyle=$
$\displaystyle
N(n)\left(A\left(\frac{i_{N(n)}(x)}{N(n)},y\right)-A\left(\frac{i_{N(n)}(x)-1}{N(n)},y\right)\right).$

This, however, implies

$\displaystyle\limsup_{n\rightarrow\infty}|K_{\mathfrak{Cb}_{N(n)}(E_{n})}(x,[0,y])-K_{\mathfrak{Cb}_{N(n)}(A)}(x,[0,y])|\leq
2\limsup_{n\rightarrow\infty}N(n)\,d_{\infty}(E_{n},A)=0\,,$

$\mathbb{P}$-almost surely, whereby the last identity is a consequence of the
definition of $N(n)$ and the fact that according to [21, Lemma 1] the
empirical copula $E_{n}$ satisfies

$d_{\infty}(E_{n},A)=O\left(\sqrt{\frac{\log(\log(n))}{n}}\right)$ (37)

with probability $1$. Altogether it follows that
$\lim_{n\rightarrow\infty}I_{n}=0$ almost surely.
Since all but at most countably many $y\in(0,1)$ fulfill (36) and weak
convergence is equivalent to convergence on a dense set, we have shown weak
convergence of $(K_{CB_{N(n)}(E_{n})}(x,\cdot))_{n\in\mathbb{N}}$ to
$(K_{A}(x,\cdot))_{n\in\mathbb{N}}$ for every $x\in\Gamma$, which completes
the proof. ∎

Having this, the following main result of this section is straightforward to
prove.

######  Theorem 4.2 (Strongly consistent estimator for
$\Lambda_{\varphi}(A)$).

Let $(X_{1},Y_{1}),(X_{2},Y_{2}),\ldots$ be a random sample from $(X,Y)$,
assume that $(X,Y)$ has continuous distribution function, and let $A$ denote
the copula of $(X,Y)$. Furthermore choose $s$ and the resolution $N(n)$ as in
Lemma 4.1. If $\varphi$ is continuous, then
$\Lambda_{\varphi}(\mathfrak{Cb}_{N(n)}(E_{n}))$ is a strongly consistent
estimator for $\Lambda_{\varphi}(A)$, i.e., with probability $1$, we have

$\lim_{n\rightarrow\infty}\Lambda_{\varphi}(\mathfrak{Cb}_{N(n)}(E_{n}))=\Lambda_{\varphi}(A).$

###### Proof.

According to (the proof of) Lemma 4.1 we have that
$(K_{CB_{N(n)}}(u,\cdot))_{n\in\mathbb{N}}$ converges weakly to
$K_{A}(u,\cdot)$ for every $u\in\Gamma$. As a direct consequence, for
$u_{1},u_{2}\in\Gamma$ it follows that

$\displaystyle\lim_{n\rightarrow\infty}|K_{CB_{N(n)(E_{n})}}(u_{1},[0,y]))-K_{CB_{N(n)(E_{n})}}(u_{2},[0,y]))|$
$\displaystyle=$ $\displaystyle|K_{A}(u_{1},[0,y]))-K_{A}(u_{2},[0,y]))|,$

hence, considering $\lambda_{2}(\Gamma\times\Gamma)=1$, using eq. (33), and
applying dominated convergence we have

$\displaystyle\lim_{n\rightarrow\infty}\int_{[0,1]^{2}}\int_{[0,1]}|K_{\mathfrak{Cb}_{N(n)(E_{n})}}(u_{1},[0,y]))-K_{\mathfrak{Cb}_{N(n)(E_{n})}}(u_{2},[0,y]))|^{p}d\lambda(y)\,d\lambda_{2}(u_{1},u_{2})$
$\displaystyle=\int_{[0,1]^{2}}\int_{[0,1]}|K_{A}(u_{1},[0,y]))-K_{A}(u_{2},[0,y]))|^{p}d\lambda(y)\,d\lambda_{2}(u_{1},u_{2})=\Lambda_{p}(A).$

∎

######  Remark 4.3.

Notice that according to Theorem 4.2 strong consistency holds in full
generality, i.e., without any smoothness restrictions for the copula $A$. We
also conjecture that the estimator
$\Lambda_{\varphi}(\mathfrak{Cb}_{N(n)}(E_{n}))$ is asymptotically normal, a
proof, however, seems still out of reach, see, e.g., [35].

######  Remark 4.4.

The checkerboard estimator can also be used for estimating
$\Lambda_{\Delta}(Y|X)$ in the more general case where the distance function
$\Delta$ is at least component-wise continuous with respect to weak
convergence and invariant with respect to strictly increasing transformations
of $Y$.

##  5 Simulation Study and Real Data Example

In this section we first study the performance of the checkerboard estimator
$\Lambda_{\varphi}(\mathfrak{Cb}_{N(n)}(E_{n}))$ from Theorem 4.2 and then
quickly discuss a real data example.

###  5.1 Simulation study

We consider the continuous setting as studied in the previous section focus on
the following choices for the convex function $\varphi$ \- notice that that
$\varphi$ is convex and strictly convex in $0=\varphi(0)$, so
$\Lambda_{\varphi}$ is a dependence measure:

  1. (i)

$\varphi(x)=|x|^{p}$ for $p\in\\{1,2,3\\}\,,$

  2. (ii)

$\varphi(x)=e^{cx}-1$ for $c\in\\{1/5,1,5\\}\,,$

  3. (iii)

$\varphi(x)=e^{|cx|}-1$ for $c\in\\{1/5,1,5\\}\,.$

As dependence structures we chose copulas $A$ from the Marshall-Olkin (MO)
family, defined by

$M_{\alpha,\beta}(u,v):=\begin{cases}u^{1-\alpha}v\,,&\text{if}\quad
u^{\alpha}\geq v^{\beta}\,,\\\ uv^{1-\beta}\,,&\text{if}\quad
u^{\alpha}<v^{\beta}\,,\end{cases}$

with $\alpha,\beta\in[0,1]$ and consider

$(\alpha,\beta)\in\\{(1,0),(1,1),(0.2,0.7),(0.3,1)\\}.$

Figures 1 to 4 illustrate the behavior of the estimator
$\Lambda_{\varphi}(\mathfrak{Cb}_{N(n)}(E_{n}))$ from Theorem 4.2 for each of
the chosen copulas $A$ and sample sizes given by

$n\in\\{10,50,100,500,1.000,5.000,10.000\\},$

where each scenario was simulated $1.000$ times. Estimation of the empirical
checkerboard copula was performed using the command ecbc from the R package
qad with the default of $s=1/2$ (see [22, Remark 3.14] for a detailed
discussion of this choice). Figures 1 to 4 can be interpreted as follows:

  * •

the measures with convex function $\varphi$ of type (ii) perform best in all
cases in terms of speed of convergence. One possible explanation of this
observation is asymmetry of $\varphi(x)=e^{cx}-1\,:$ The number of distinct
values over which the integral runs is potentially twice as large and might
therefore speed up convergence. As a drawback, however, for these functions,
$\Lambda_{\varphi}(Y|X)$ often attains small values, see Figures 3 and 4,
which, however should not be confused with independence as considered in
Figure 1.

  * •

$\Lambda_{\varphi}(Y|X)$ is decreasing in the parameters $p$ and $c$ (compare
with Example 3). Consequently, in the case of complete dependence, the
measures with parameter $p=1$ and $c=1/5$ perform best because deviations are
less penalized by the shape of the function $\varphi$ than by the other cases.
In contrast to that, if the parameters $p$ and $c$ are large, then the measure
$\Lambda_{\varphi}$ is more sensitive for detecting independence.

  * •

the $L^{2}$-version $\varphi(x)=x^{2}$ as well as the asymmetric version
$\varphi(x)=e^{x}-1$ perform best over all considered cases.

Figure 1: Simulation Results for the MO copula with parameters $\alpha=1$ and
$\beta=0$, i.e., for the independence copula. The top left panel depicts a
random sample of size $n=5.000$ from this distribution. The other panels
contain boxplots of the estimations
$\Lambda_{\varphi}(\mathfrak{Cb}_{N(n)}(E_{n}))$ according to Theorem 4.2
based on $1,000$ runs for each sample size $n$ and for each convex function
$\varphi$. The dashed lines represent the true values of
$\Lambda_{\varphi}(A)$. Figure 2: Simulation Results for the MO Copula with
parameters $\alpha=1$ and $\beta=1$, i.e., for the case of perfect positive
dependence.The top left panel depicts a random sample of size $n=5.000$ from
this distribution. The other panels contain boxplots of the estimations
$\Lambda_{\varphi}(\mathfrak{Cb}_{N(n)}(E_{n}))$ according to Theorem 4.2
based on $1,000$ runs for each sample size $n$ and for each convex function
$\varphi$. The dashed lines represent the true values of
$\Lambda_{\varphi}(A)$. Figure 3: Simulation Results for the MO Copula with
parameters $\alpha=0.2$ and $\beta=0.7$. The top left panel depicts a random
sample of size $n=5.000$ from this distribution. The other panels contain
boxplots of the estimations $\Lambda_{\varphi}(\mathfrak{Cb}_{N(n)}(E_{n}))$
according to Theorem 4.2 based on $1,000$ runs for each sample size $n$ and
for each convex function $\varphi$. The dashed lines represent the true values
of $\Lambda_{\varphi}(A)$. Figure 4: Simulation Results for the MO Copula with
$\alpha=0.3$ and $\beta=1$. The top left panel depicts a random sample of size
$n=5.000$ from this distribution. The other panels contain boxplots of the
estimations $\Lambda_{\varphi}(\mathfrak{Cb}_{N(n)}(E_{n}))$ according to
Theorem 4.2 based on $1,000$ runs for each sample size $n$ and for each convex
function $\varphi$. The dashed lines represent the true values of
$\Lambda_{\varphi}(A)$.

###  5.2 Real Data Example

As a real data example we consider a dataset containing gathered by [30] and
available online at PhysioNet.org, see [17]. The dataset contains, among other
variables, $101$ complete observations of pregnant women’s age, BMI, blood
pressure, blood glucose level and thickness of visceral adipose tissue as well
as the birth weight of the child. We choose the birth weight of the child as
the endogenous variable $Y$ and each of the other variables as the exogenous
one. Figure 5 illustrates the empirical checkerboard copula
$\mathfrak{Cb}_{N(n)}(E_{n}))$ for BMI as exogenous variable. Table 1
summarizes the obtained values of $\Lambda_{\varphi}(Y|X)$ for all exogenous
variables. Interpretation of the different values is not straightforward since
they depend on the shape of $\varphi$ (unless they are exactly $0$ or $1$).
Due to the small sample size the calculated estimates might still differ from
the underlying true value. Surprisingly, however, when sorting the variables
by their estimated $\Lambda_{\varphi}(Y|X)$ value, the order is the same for
all convex functions $\varphi$, i.e., for all choices of $\varphi$ the
variable importance (w.r.t. $Y$) is the same.

Figure 5: The left panel depicts the Empirical Checkerboard Copula $\mathfrak{Cb}_{N(n)}(E_{n}))$ with resolution $N=9$ and together with the pseudo-observations (a.k.a. psuedo-observations) for Birth Weight ($y$-axis) and BMI of the mother ($x$-axis). The right plot contains the resulting conditional distribution functions $F_{Y|X=x}$ for the $9$ different BMI regions corresponding to the $9$ vertical stripes on the left plot. $\Lambda_{\Delta}(Y|X)$ is computed by taking the average pairwise distances $\Delta(F_{i},F_{j}),\,i,j\in\\{1,\ldots,9\\}$ over these $9$ ECDFs and subsequently dividing by $\alpha_{\Delta}$. Table 1: Values of $\Lambda_{\varphi}(Y|X)$ for various choices of the exogenous variable $X$ and for different convex functions $\varphi$. In each of the cases the endogenous variable $Y$ is the birth weight. | $\varphi(x)$
---|---
Exogenous Variable | $|x|$ | $e^{|x|}-1$ | $e^{x}-1$
Diastolic BP | $0.354$ | $0.231$ | $0.072$
BMI | $0.326$ | $0.210$ | $0.060$
Visceral Adipose Tissue | $0.304$ | $0.195$ | $0.052$
Systolic BP | $0.291$ | $0.186$ | $0.049$
Blood Glucose | $0.214$ | $0.134$ | $0.027$
Age | $0.137$ | $0.083$ | $0.012$

Funding
The authors gratefully acknowledge the support of the WISS 2025 project ‘IDA-
lab Salzburg’ (20204-WISS/225/197-2019 and 20102-F1901166-KZP). The first and
the third author further acknowledge the support of the Austrian Science Fund
(FWF) project P 36155-N _ReDim: Quantifying Dependence via Dimension
Reduction_.

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