# Measures of independence and functional dependence

Peter J. Bickel
University of California, Berkeley

###### Abstract

We follow up on Shi et al’s (2020) and Cao’s and my (2020) work on the local
power of a new test for independence, Chatterjee (2019), and its relation to
the local power properties of classical tests. We show quite generally that
for testing independence with local alternatives either Chatterjee’s rank test
has no power or it may be misleading: The Blum,Kiefer,Rosenblatt and other
omnibus classical rank tests do have some local power in any direction other
than those where significant results may be misleading. We also suggest
methods of selective inference in independence testing. Chatterjee’s
statistics like Renyi’s Rényi (1959) also identified functional dependence. We
exhibit statistics which have better power properties than Chatterjee’s but
also identify functional dependence.

##  1 Introduction

In a remarkable paper Chatterjee (2020), Sourav Chatterjee proposed a new
coefficient of correlation based on an i.i.d. sample
$(X_{i},Y_{i}),i=1,\ldots,n$. Assuming there are no ties among the $X_{i}$’s
and $Y_{i}$’s (see Chatterjee (2020) for the definition in the general case),
the correlation is defined as

$\widehat{C}_{n}(X,Y):=1-\frac{3\sum_{i=1}^{n-1}|r_{i+1}-r_{i}|}{n^{2}-1},$
(1.1)

where the $r_{i}$ are defined as follows. First, sort $X_{(1)}\leq\cdots\leq
X_{(n)}$, and for each $i$ let $Y_{(i)}$ be the $Y$ sample corresponding to
$X_{(i)}$. Then $r_{i}$ is defined as the rank of $Y_{(i)}$, i.e. the number
of $j$ such that $Y_{j}\leq Y_{(i)}$. Chatterjee showed that as
$n\rightarrow\infty$, $\widehat{C}_{n}$ a.s. converges to the population
measure

$\displaystyle C(X,Y):=\frac{\int\mathrm{Var}(\mathbb{E}[\mathbbm{1}(Y\geq
t)~{}|~{}X])d\mu(t)}{\int\mathrm{Var}(\mathbbm{1}(Y\geq t))d\mu(t)}$

$\displaystyle=1-\frac{\int\mathbb{E}[\mathrm{Var}(\mathbbm{1}(Y\geq
t)~{}|~{}X)]d\mu(t)}{\int\mathrm{Var}(\mathbbm{1}(Y\geq t))d\mu(t)},$ (1.2)

where $\mu$ is the law of $Y$. Here $Y$ is assumed to not be constant. In the
case where $X,Y$ are continuously distributed, this measure was introduced by
Dette et al. Dette et al. (2013). The measure $C$ has a number of interesting
properties:

  1. A)

$0\leq C\leq 1$.

  2. B)

$C=0$ if and only if $X$ and $Y$ are independent.

  3. C)

$C=1$ if and only if $Y=h(X)$ a.s. for some measurable function
$h:\mathbb{R}\rightarrow\mathbb{R}$.

  4. D)

$C$ is asymmetric, but can be easily symmetrized to

$C^{*}(X,Y):=\max(C(X,Y),C(Y,X)),$

which clearly satisfies $C^{*}=1$ if and only if $X$ is a function of $Y$ or
$Y$ is a function of $X$ (or both).

  5. E)

$C$ is invariant under strictly increasing transformations of $X$ and $Y$
separately.

This measure is akin to the Rényi correlation (also commonly called the
maximal correlation), which we shall denote $R$ or $R(X,Y)$, and is defined as
the maximum Pearson correlation between all pairs of $L^{2}$ functions of $X$
and $Y$ respectively. $R$ may be computed as the square root of the maximal
eigenvalue of a compact self adjoint operator,

$T:L^{2}_{0}(X)\rightarrow L^{2}_{0}(X),$

(or $L^{2}_{0}(Y)\rightarrow L^{2}_{0}(Y)$ with appropriate changes), where
$L^{2}_{0}(X)$ is the subspace of $L^{2}(X)$ consisting of mean zero random
variables. The operator $T$ is given by

$T(f(X)):=\mathbb{E}[\mathbb{E}[f(X)~{}|~{}Y]~{}|~{}X].$ (1.3)

The Rényi correlation is well known to have properties $A,B,$ and $C$, but it
is symmetric, and

  1. Cenumi)

$R=1$ if and only if $g(Y)=h(X)$ for some functions $g$ and $h$, with $g(X)\in
L^{2}(X)$ and $h(Y)\in L^{2}(Y)$.

An extensive account of the history and computation of estimates of R, and
other properties of $R$ may be found for instance in Bickel et al. (1998);
Rényi (1959).

It is clear that estimators of $R$, generically call them $\widehat{R}$,
involve implicitly estimation of the conditional distribution of $Y$ given
$X$, and conversely. Thus, except in the finite discrete case, they require
assumptions on smoothness of the joint distribution of X and Y for convergence
to $R$. In testing the hypothesis $H:\textit{X independent of Y}$ versus
contiguous alternatives, $\widehat{R}$ has no power, i.e. has asymptotic power
equal to the level of significance.

Perhaps surprisingly, it has been shown independently by Shi et al.(2020) Shi
et al. (2021) and Cao and Bickel Cao and Bickel (2020) (2020) that the test
based on $\widehat{C}_{n}$ and the related one based on Dette et al’s
statistic have the same poor behaviour as tests of independence in many copula
models.

In this paper, we show quite generally that Chatterjee’s test has no power
(power $\alpha$) against local parametric alternatives (those converging to
$H$ at rate $O(\frac{1}{\sqrt{n}})$ ) other than a special set of parametric
families. Those alternatives can yield significant results but are equivalent
in a strong sense to models where $X$ and $Y$ are always independent, varying
only in their marginal distributions.

We go on to show that this behaviour is to be expected from a large family of
rank tests which use the ranks of $X$’s belonging to adjacent $X$’s. Or, put
another way, these tests look at departures from independence from a very
local point of views in $(X,Y)$ space.

Following an approach of Bickel, Ritov,Stoker (2006) Bickel et al. (2006), we
also introduce a class of simple tests, computable in $O(n)$ steps, which have
power tailored to the complexity of the alternative, and develop this into an
approach to selective inference.

In section 3, we discuss functional dependence and a principle underlying both
Renyi’s and Chatterjee’s statistics, deriving some new procedures and relating
these to classical ones in a general context, including multivariate and
discrete ones.

Finally, in Section 4, we discuss the various ways by which parameters
satisfying both of Chatterjee’s properties B and C can be constructed with
satisfactory behaviour under both regimes.

##  2 Tests of independence

###  2.1 Definitions

There has recently been a surge of interest in such tests. From the point of
view of minimax analysis (Kim et al. (2020)), local power analysis (Shi et al.
(2021); Weihs et al. (2018), Cao and Bickel (2020), many types of statistics
(Genest et al. (2019); Lin and Han (2021); Székely et al. (2007)) have been
proposed.

In addition to Chatterjee’s and Dette et al’s $C$, there are a number of well
known population parameters corresponding to test statistics based on ranks or
more generally invariant under monotone transformations of coordinates which
are consistent against essentially all fixed alternatives to $H:$ $X$
independent of $Y$. Several prominent ones are due to Hoeffding(1948),
Blum,Kiefer,Rosenblatt(1961) Blum et al. (1961), Bergsma and Dassios(2014)
Bergsma and Dassios (2014), and more recently Weihs, Drton, Meinshausen(2018)
Weihs et al. (2018) and many others (Kim et al(2020) Kim et al. (2020)). In
addition there are even more classical parameters corresponding to directed
statistics such as Spearman’s $\rho$ and Kendall’s $\tau$. All of these
statistics, $T_{n}$, can be related to population parameters $T(F)$ where $F$
ranges over all bivariate distributions, represented as cdfs, and which have
the property that $T(F)=0$ if $X$ and $Y$ are independent. This is Weihs et
als Weihs et al. (2018), property I. The statistics $T_{n}$ are typically, but
not necessarily, obtained by plug in but will be identified by consistency to
the population parameters as $n\to\infty$. Plug in means,

$\displaystyle T_{n}\equiv T(\widehat{F})$

,where $\widehat{F}$ is the empirical cdf of the sample $(X_{i},Y_{i})$ from
$F$.

The class of parameters we consider are functions, not just of $F$, but of
$C$, defined as the cdf of the vector $(U,V)$ with $U=F_{1}(X)$, $V=F_{2}(Y)$,
where $F_{1},F_{2}$ are the cdf of $X,Y$ respectively, and $U$ and $V$ have
corresponding marginals $C_{1}$ and $C_{2}$. If $F_{1}$ and $F_{2}$ are
continuous, C is a copula and $C_{1},C_{2}$ are both $U(0,1)$. In general,

$\displaystyle C(u,v)=F(Q_{1}(u),Q_{2}(v))$

,where $Q_{j}(.)$ is the inverse of $F_{j}$ and the inverse is defined for any
cdf $F$ by,

$\displaystyle Q(u)=\inf\\{x:F(x)\geq u\\}$

The corresponding $T_{n}$ will depend on the empirical cdf of the empirical
probability integral transforms, $\widehat{F_{1}}(X),\widehat{F_{2}}(Y))$.
This object, we shall call $\widehat{C}$, is referred to as the empirical
copula Tsukahara (2005). Thus,

$\displaystyle\widehat{C}(u,v)=\widehat{F}(\widehat{Q_{1}}(u),\widehat{Q_{2}}(v))$
(2.1)

where $\widehat{Q_{j}}$ are just the quantile functions of $\widehat{F_{j}}$.

Up to this point we have considered all $F$, continuous, discrete or both. For
testing $H$, the assumption that $F_{j},j=1,2$ are continuous, manifested by
no ties in the $X$’s or $Y$’s, simplifies matters considerably, as we have
partly noted. The distribution of $T(\widehat{C})$ under $H$ does not depend
on F. It is always a function of $C$, the cdf of the $(U_{i},V_{i}),i=1,...,n$
only.

If $\widehat{C}$ is the empirical cdf of the vectors of ranks normalized by n,
being a function of $\widehat{C}$ simply means that the $T_{n}$ are ordinary
rank statistics. That is, $T_{n}=T_{n}(S_{(1)},…,S_{(n)})$ where $S_{(j)}$ is
the rank of the $Y$ corresponding to the order statistic $X_{(j)}$. All the
classical statistics for testing independence, as well as the more recent ones
of Weihs et al., Bergsma et al., as well as Chatterjee and Dette et al., are
functions of the empirical copula.

As examples, we consider,

1) The Blum Kiefer-Rosenblatt parameter

$\displaystyle T_{BKR}(F)$
$\displaystyle\equiv\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}(F(x,y)-F_{1}(x)F_{2}(y))^{2}dF_{1}(x)dF_{2}(y)$
$\displaystyle=\int_{0}^{1}\int_{0}^{1}(C(u,v)-C_{1}(u)C_{2}(v))^{2}dudv$
$\displaystyle=\int_{0}^{1}\int_{0}^{1}(C(u,v)-uv)^{2}dudv$ (2.2)

in the copula case.

2) The Spearman correlation parameter

$\displaystyle T_{S}(F)$ $\displaystyle\equiv corr(F_{1}(x),F_{2}(Y))$ (2.3)
$\displaystyle=\frac{\int_{0}^{1}\int_{0}^{1}(uv-\mu^{*}_{U}\mu^{*}_{V})dC(u,v)}{\sigma^{*}_{U}\sigma^{*}_{V}}$

where $\mu^{*}_{U},\mu^{*}_{V},\sigma^{*}_{U},\sigma^{*}_{V}$ are the means
and variances of $U\equiv F_{1}(X),V\equiv F_{2}(Y)$.

If C is a copula, $T_{S}(F)=12[\int_{0}^{1}\int_{0}^{1}uvdC(u,v)-\frac{1}{4}]$

3) The sup norm parameter, we call $T_{K}$ for Kolmogorov, introduced in Blum
et al. (1961).

$\displaystyle T_{K}(F)$
$\displaystyle\equiv\sup\\{|F(x,y)-F_{1}(x)F_{2}(y)|:x,y\\}$
$\displaystyle=\sup\\{C(u,v)-C_{1}(u)C_{2}(v):u,v\in[0,1]^{2}\\}$

The familiar corresponding statistics are obtained up to $O_{p}(\frac{1}{n})$
by replacing $F$ by $\widehat{F}$ or $C$ by $\widehat{C}$. If $C$ is a copula,
parameters and statistics simplify since $C_{1}(v)=C_{2}(v)=v$, and statistics
are functions of the normalized ranks.

The statistics can then be written in terms of $r_{i}$ defined for (1.1) as,

$\displaystyle\widehat{T}_{S}$
$\displaystyle=\frac{12}{n}\sum_{i=1}^{n}(\frac{ir_{i}}{n^{2}}-\frac{1}{4})+O_{p}(\frac{1}{n})$
$\displaystyle\widehat{T}_{K}$
$\displaystyle=max\\{|\frac{1}{n}\sum_{i=1}^{j}(1(r_{i}\leq
k)-\frac{jk}{n^{2}})|:1\leq j,k\leq n\\}+O_{p}(\frac{1}{n})$
$\displaystyle\widehat{T}_{BKR}$
$\displaystyle=\frac{1}{n^{2}}\sum_{i,j=1}^{n}A_{ij}B_{ij}-\frac{1}{2n}\sum_{i=1}^{n}(\frac{ir_{i}}{n})^{2}-\frac{7}{18}+O_{p}(\frac{1}{n})$

where $A_{ij}=\frac{i}{n}\wedge\frac{j}{n}$,
$B_{ij}=\frac{r_{i}}{n}\wedge\frac{r_{j}}{n}$.

Both $\widehat{T}_{BKR}$ and $\widehat{T}_{K}$ require $O(n^{2})$ operations.
$\widehat{T}_{S}$ requires $O(n\log n)$ operations.

There are, of course, many other parameters measuring independence, discussed
in Tsukahara (2005), Weihs et al (2018) among others. The corresponding
statistics are typically plug in. Most, such as
$\widehat{T}_{S},\widehat{C}_{n}$, are asymptotically Gaussian, and linear, in
the sense that,

$\displaystyle T(\widehat{F})$
$\displaystyle=T(F)+\frac{1}{n}\sum_{i=1}^{n}{\Psi(X_{i},Y_{i},F)}+o_{p}(n^{-\frac{1}{2}})$
(2.4)

with influence function $\Psi\in L_{2}(F),E_{F}\Psi(X,Y,F)=0$.

Some are not, for example $\widehat{T}_{BKR}$ and $\widehat{T}_{K}$, given by,

$\displaystyle\widehat{T}_{K}\equiv
sup\\{|\widehat{C}(u,v)-\widehat{C_{1}}(u)\widehat{C_{2}}(v)|:u,v\\}.$

The limiting behavior of these statistics is formally derivable using the
stochastic process
$\sqrt{n}(\widehat{C}-C)$ which we are about to introduce. In some cases, such
as the Weihs et al statistics. $U$ statistic theory provides much simpler
proofs. By (1.2), the Chatterjee-Dette et al parameter, which we shall call
$T_{C}(F)$ ,can also be expressed in terms of $F$,

$\displaystyle T_{C}(F)$ $\displaystyle=\frac{E_{F}\bigg{(}Var_{F}(1(Y\leq
Y^{\prime})|X)\bigg{)}}{Var_{F}(1(Y\leq Y^{\prime}))}$ (2.5)

where $(X,Y)\sim F$ and $Y^{\prime}$ is independent of $(X,Y)$ and distributed
according to $F_{2}(.)$.

We return to this definition in section 3.4. $C$ can replace $F$ in this
definition. However, the corresponding statistics in Chatterjee (2020) and
Dette et al. (2013) are not plug ins. Our formulae so far are all valid for
general $F$, $\widehat{F}$. For the rest of this section, we focus on the
copula case, F continuous. We return to the general case at the end of the
section, in a brief discussion.

###  2.2 Stochastic Processes associated with the hypothesis $H$ of
independence

The process,

$\displaystyle T_{n}(u,v)=\sqrt{n}\bigg{(}\widehat{C}(u,v)-C(u,v)\bigg{)}$
(2.6)

has been heavily studied, (Genest et al. (2019),Tsukahara (2005), for
example). In general, its behavior requires delicate conditions because of its
dependence on $\widehat{Q}_{j}$. However, for the situation we focus on, $X,Y$
independent and $F$ continuous, the following special case of the general
theory characterizing $T_{n}(.,.)$ is simple.

Let $E_{n},E_{n1}$ and $E_{n2}$ be the empirical process of
$\\{(U_{i},V_{i})\\},\\{U_{i}\\}$ and $\\{V_{i}\\}$ respectively where
$U_{i}=F_{1}(X_{i}),V_{i}=F_{2}(Y_{i})$ are i.i,d. $U(0,1)$. Then, $C(u,v)=uv$
and $\widehat{C}(.,.)$ is the same as the empirical copula of the
$(U_{i},V_{i})$. Thus, if we simplify notation and set $F_{1},F_{2}$ to the
identity,

$\displaystyle\sqrt{n}\bigg{(}\widehat{C}(u,v)-C(u,v)\bigg{)}=E_{n}(\widehat{Q}_{1}(u),\widehat{Q}_{2}(v))+\sqrt{n}(\widehat{Q}_{1}(u)\widehat{Q}_{2}(v)-uv).$
(2.7)

By classical empirical process theory and Bahadur Bahadur (1966),

$\displaystyle\sqrt{n}(\widehat{Q}_{j}(u)-u)=-E_{nj}(u)+o_{p}(1),j=1,2$ (2.8)

and,

$\displaystyle E_{n}(Q_{1}(u),Q_{2}(v))=E_{n}(u,v)+o_{p}(1).$ (2.9)

Note that $o_{p}(1)$ and similar expressions denote the order of functions in
the $L_{\infty}$ norm.

By the delta method,

$\displaystyle\sqrt{n}(\widehat{Q}_{1}(u)\widehat{Q}_{2}(v)-uv)=u\sqrt{n}(\widehat{Q}_{2}(v)-v)+v\sqrt{n}(\widehat{Q}_{1}(u)-u)$
(2.10) $\displaystyle=-uE_{n2}(v)-vE_{n1}(u)+O_{p}(n^{-\frac{1}{2}})$ (2.11)

by (2.8).

We conclude that,

######  Proposition 2.1.

If $X$ and $Y$ are independent and $F$ is continuous,
$T_{n}(u,v)=E_{n}(u,v)-uE_{n2}(v)-vE_{n1}(u)+o_{p}(1)$

This is a special case of Proposition 1 of Tsukahara (2005)Tsukahara (2005).

We now claim that, under the same conditions,

$\displaystyle\widetilde{\Lambda}_{n}(u,v)\equiv
uE_{n2}(v)+vE_{n1}(u)=\Pi(E_{n}(u,v)|\Lambda_{n})$ (2.12)

Here $\Pi(.|\Lambda_{n})$ denotes projection in $L_{2}(U_{i},V_{i}:i=1,...,n)$
on the linear subspace $\Lambda_{n}$ given by,

$\displaystyle\Lambda_{n}=\\{\sum_{i=1}^{n}{(f_{i}(U_{i})+g_{i}(V_{i})}:Ef_{i}^{2}(U)<\infty,Eg_{j}^{2}(V)<\infty,Ef_{i}(U)=Eg_{j}(V)=0\textit{
for all }i,j\\}$ (2.13)

This follows from the well known observations:

  1. 1.

If $h(U,V)\in L_{2}(U,V)$, the projection of $h$ on the space of all functions
of $U$ is $E(h(U,V)|U)$ and similarly for functions of $V$

  2. 2.

The projection operator is additive on a sum of orthogonal subspaces.

Thus, our claim is equivalent to,

$\displaystyle E(1(U\leq u,V\leq v)|V)-uv=u(1(V\leq v)-v)$

and similarly for $U$, and (2.12) follows.

Denote the process of (2.12) by $\Lambda_{n}(u,v)$. Let $D[0,1]^{2}$ be
defined as in Billingsley Billingsley (1968).

######  Proposition 2.2.

Under the conditions of Proposition 2.1, let $g,h:D[0,1]^{2}\to
D[0,1]^{2},S_{n}\equiv g(T_{n}(.,.)),V_{n}\equiv
h(\widetilde{\Lambda}_{n}(.,.))$. $g,h$ measurable on $D[0,1]^{2}$.
Then, if $(S_{n},V_{n})\xrightarrow{d}(S,V)$, $S$ and $V$ are independent.

$\mathbb{Proof:}$ From classical empirical process theory,
$(T_{n}(.,.),\widetilde{\Lambda}_{n}(.,.))$ converge jointly weakly to jointly
Gaussian processes with the same covariance structure as
$E_{n}(u,v)-uE_{n2}(v)-vE_{n1}(u)$ and $uE_{n2}(v)+vE_{n1}(u)$, which we shall
call $T^{*}$ and $\Lambda$. We shall denote the limit of $E_{n}(.)$ by $E(.)$
and those of $E_{nj}(.)$ by $E_{j}(.)$. Note that we can think of these limit
processes as a.s. limits in the $l_{\infty}$ norm. By (2.12),
$\widetilde{\Lambda}_{n}$ and $E_{n}-\widetilde{\Lambda}_{n}$ are orthogonal.
Their Gaussian limits must then be independent.∎

Formulae for the covariances functions of $\widetilde{\Lambda}_{n}$ and
$T_{n}$ and hence of $T^{*}$ follow. If
$\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}}\equiv(u_{1},u_{2})$,
$\hbox{\oalign{$\bm{v}$\crcr$\scriptscriptstyle\bm{\sim}$}}\equiv(v_{1},v_{2})$,

$\displaystyle
cov(T^{*}(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}}),T^{*}(\hbox{\oalign{$\bm{v}$\crcr$\scriptscriptstyle\bm{\sim}$}}))$
$\displaystyle=cov(E_{n}(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}}),E_{n}(\hbox{\oalign{$\bm{v}$\crcr$\scriptscriptstyle\bm{\sim}$}}))-cov(\widetilde{\Lambda}_{n}(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}}),\widetilde{\Lambda}_{n}(\hbox{\oalign{$\bm{v}$\crcr$\scriptscriptstyle\bm{\sim}$}}))$
(2.26) $\displaystyle=(u_{1}\wedge u_{2})(v_{1}\wedge
v_{2})-u_{1}u_{2}v_{1}v_{2}-u_{1}v_{1}[(u_{2}\wedge
v_{2})-u_{2}v_{2}]-u_{2}v_{2}[(u_{1}\wedge v_{1}-u_{1}v_{1})]$
$\displaystyle=(u_{1}\wedge v_{1}-u_{1}v_{1})(u_{2}\wedge v_{2}-u_{2}v_{2})$
(2.27)

as noted by Tsukahara. We can represent

$\displaystyle\Lambda(u,v)=uW^{\circ}(v)+vW^{\circ}(u)$

where $W^{\circ}$ is a Brownian Bridge.

We proceed with the implication of this result for local power.

###  2.3 Local power against H

####  2.3.1 Le Cam Theory

We make the following assumptions,following Le Cam and Yang (1990) Cam and
Yang (1990),

  1. I

We begin with a model $\mathcal{M}=\\{h_{\theta}(x,y):|\theta|<1\\}$, where
$h_{\theta}(x,y)$ is a joint density with respect to Lebesgue measure, and
$h_{0}(x,y)=f_{0}(x)g_{0}(y)$ (independence).

  2. II

Suppose, the score function,

$\dot{\ell}_{0}(x,y):=\frac{\partial}{\partial\theta}\log
h_{\theta}(x,y)\bigg{|}_{\theta=0}$

exists, and suppose the family $\\{h_{\theta}(x,y)\\}_{|\theta|<1}$ is
quadratic mean differentiable (QMD) at $\theta=0$ with score function
$\dot{\ell}_{0}$. That is, we have

$\int\int\big{(}\sqrt{h_{\theta}(x,y)}-\sqrt{h_{0}(x,y)}-(1/2)\theta\dot{\ell}_{0}(x,y)\sqrt{h_{0}(x,y)}\big{)}^{2}dxdy=o(\theta^{2}).$

Moreover, assume $\mathbb{E}_{0}\dot{\ell}_{0}(X,Y)^{2}>0$ (the assumption of
quadratic mean differentiability implies
$\mathbb{E}_{0}[\dot{\ell}_{0}(X,Y)^{2}]<\infty$ and
$\mathbb{E}_{0}\dot{\ell}_{0}(X,Y)=0$).

For $\theta\in\mathbb{R}$ let $P_{n,\theta}$ be the product measure on
$([0,1]\times[0,1])^{n}$ with density $\prod_{i=1}^{n}h_{\theta}(x,y)$. Take
$t\in\mathbb{R}$ and let $\theta_{n}:=t/\sqrt{n}$. By Le Cam’s Theorem (see
e.g. (van der Vaart, 1998, Theorem 7.2 and Example 6.5)), under our
assumptions $\\{P_{n,\theta_{n}}\\}_{n\geq 1}$ is contiguous with respect to
$\\{P_{n,0}\\}_{n\geq 1}$. That is if $Q_{n}$ is a function of
$((X_{i},Y_{i}),1\leq i\leq n)$, then

$Q_{n}\stackrel{{\scriptstyle P_{n,0}}}{{\longrightarrow}}0\text{ implies
}Q_{n}\stackrel{{\scriptstyle P_{n,\theta_{n}}}}{{\longrightarrow}}0.$

For our purposes, more importantly, Le Cam’s third lemma holds (see e.g. (van
der Vaart, 1998, Theorem 7.2 and Example 6.7)), stating that if,

$\displaystyle\bigg{(}\sqrt{n}Q_{n},\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\dot{\ell}_{0}(X_{i},Y_{i})\bigg{)}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}N\bigg{(}\begin{pmatrix}0\\\
0\end{pmatrix},\begin{pmatrix}\sigma^{2}&\rho\sigma\tau_{0}\\\
\rho\sigma\tau_{0}&\tau_{0}^{2}\end{pmatrix}\bigg{)}$ (2.28)

under $\\{P_{n,0}\\}_{n\geq 1}$, where
$\tau_{0}^{2}:=\mathbb{E}_{0}[\dot{\ell}_{0}(X,Y)^{2}]$ is the Fisher
information, then under $\\{P_{n,\theta_{n}}\\}_{n\geq 1}$, we have

$\sqrt{n}Q_{n}\stackrel{{\scriptstyle d}}{{\longrightarrow}}N(tc,\sigma^{2}),$

where $c:=\rho\sigma\tau_{0}$ is the asymptotic covariance (under
$\\{P_{n,0}\\}_{n\geq 1}$) between the statistic $T_{n}$ and the score
statistic.

Assuming (2.28) holds under $\\{P_{n,0}\\}_{n\geq 1}$, $\sqrt{n}T_{n}/\sigma$
can be viewed as a test statistic for the hypothesis of independence, while

$\displaystyle
L_{n}:=\frac{1}{\tau_{0}\sqrt{n}}\sum_{i=1}^{n}\dot{\ell}_{0}(X_{i},Y_{i})$
(2.29)

can be viewed as the asymptotically optimal test statistic for the family
$\\{h_{\theta}(x,y)\\}_{|\theta|<1}$. The Pitman efficiency (see e.g. (van der
Vaart, 1998, Chapter 8)) of the first test to the second is by (2.16),

$e(Q_{n})=\rho^{2}.$

Note that $e(L_{n})=1$.

$\mathbb{Remark:}$ 1) Note that if $\dot{\ell}_{0}(x,y)$ exists as a pointwise
derivative and is continuous in $L_{2}$ norm at $\theta=0$, QMD follows.

2) Let $U\equiv F_{1}(X),V\equiv F_{2}(Y)$ where $F_{1},F_{2}$ are the cdf of
$X$ and $Y$ respectively if $\theta=0$. If we equivalently view $\mathcal{M}$
as the model of distributions of $(U,V)$, it is easy to see that the
corresponding score function is unchanged as a random variable and is given by
$\dot{l}_{0}(F_{1}^{-1}(U),F_{2}^{-1}(V))$. In what follows, we shall usually
WLOG assume $F_{1},F_{2}$ are the identity and hence $X=U$, $Y=V$ both
$\mathcal{U}(0,1)$ under $H$. We equivalently use $\dot{l}_{0}(U,V)$ for
$\dot{l}_{0}(X,Y)$.

We will use a slight generalization of the third lemma, which is evident once
we note that contiguity preserves tightness.

######  Lemma 2.1.

Let $M_{n}(.)$ be a stochastic process taking values in $D^{k}\equiv
D([0,1]^{k})$ (D as defined in Billingsley Billingsley (1968) ). Suppose under
$\\{P_{n,0}\\}_{n\geq 1}$,

$\displaystyle\bigg{(}\sqrt{n}M_{n}(.),\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\dot{\ell}_{0}(X_{i},Y_{i})\bigg{)}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\bigg{(}M(.),Z\bigg{)}$

where $\stackrel{{\scriptstyle d}}{{\longrightarrow}}$ indicates weak
convergence and $\bigg{(}M(.),Z\bigg{)}$ is a Gaussian process,
$\mathbb{E}M(.)=0$, and $cov\bigg{(}M(s),M(t)\bigg{)}=\sigma(s,t)\textit{ for
}\sigma:D^{k}\times D^{k}{\longrightarrow}\mathbb{R}$.

Let,

$\displaystyle C(s)=cov\bigg{(}M(s),Z\bigg{)}$

Then, under $\\{P_{n,\theta_{n}}\\}$,

$\displaystyle\sqrt{n}M_{n}(.)\stackrel{{\scriptstyle
d}}{{\longrightarrow}}M(.)+tC(.)$

####  2.3.2 Some Definitions

Suppose $T_{n}$ is a rank statistic. and a critical region is $[T_{n}\geq
t_{n}]$. A test based on $T_{n}$ has _asymptotic level_ $\alpha$ iff

$\displaystyle\lim_{n}P_{0}[T_{n}\geq t_{n}]=\alpha.$

A level $\alpha$ test is _asymptotically unbiased_ iff for all regular one
dimensional parametric submodels with efficient score function $l_{0}^{*}$ and
$|\theta_{n}|\leq\frac{M}{\sqrt{n}},M<\infty$,

$\displaystyle\liminf P_{\theta_{n}}[T_{n}\geq t_{n}]>\alpha.$

It has _”no power”_ asymptotically if for all submodels as above,
$\frac{\epsilon}{\sqrt{n}}\leq|\theta_{n}|\leq\frac{M}{\sqrt{n}},$ all
$\epsilon>0,M<\infty$,

$\displaystyle\liminf P_{\theta_{n}}[T_{n}\geq t_{n}]=\alpha.$ (2.30)

####  2.3.3 Local power behaviour of $\widehat{C}_{n}$

We apply our machinery to $\widehat{C}_{n}$.

We note, if $H$ is true and $F$ is continuous, then by Angus(1995), see also
Shi et al (2021) and Cao and Bickel (2021), we have

$\displaystyle\widehat{C}_{n}(X,Y)=\frac{1}{n}\sum_{i=1}^{n}{(\frac{2}{3}-|V_{[i]}-V_{[i+1]}|-2V_{[i]}(1-V_{[i]}))}+o_{p}(n^{-\frac{1}{2}})$
(2.31)

where we define $V_{[i]}$ by, $V_{[i]}=V_{j}$ iff $Rank(U_{j})=i$, and
$U_{i}\equiv F_{1}(X_{i}),V_{i}\equiv F_{2}(Y_{i})$.

We claim that,

######  Proposition 2.3.

If $\dot{l}_{0}$ is as defined in (2.16) and

$\displaystyle E_{0}(\dot{l}_{0}(X,Y)|X)=E_{0}(\dot{l}_{0}(X,Y)|Y)=0$ (2.32)

or equivalently,

$\displaystyle\dot{l}_{0}(U_{1},V_{1})\in\Lambda_{1}^{\perp}$ (2.33)

then $\widehat{C}_{n}$ has no power.

Proof:  This proposition follows from the more general Theorem 2.3 and the
following.

Let $\mathcal{M}$ be a model as defined in Section 2.3.1 with density
$h_{\theta}(.,.)$ and score function $\dot{l}_{0}(X,Y)$. Without loss of
generality we let $F_{1}=F_{2}=Identity$ and write $\dot{l}_{0}(U,V)$.
Decompose $\dot{l}_{0}$ as,

$\displaystyle\dot{l}_{0}(U,V)=\dot{l}_{00}(U,V)+\dot{l}_{01}(U,V)$

where $\dot{l}_{00}\in\Lambda_{1}$ and $\dot{l}_{01}\in\Lambda_{1}^{\perp}$,
where $\Lambda_{1}$ is defined in (2.13).

For a statistic $Q_{n}$, satisfying the hypothesis of Section 2.3.1, let,

$\displaystyle\lambda_{0}(Q_{n})$ $\displaystyle\equiv
lim_{n}cov_{0}(\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\dot{l}_{00}(U_{i},V_{i}),Q_{n})$
$\displaystyle\lambda_{1}(Q_{n})$ $\displaystyle\equiv
lim_{n}cov_{0}(\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\dot{l}_{01}(U_{i},V_{i}),Q_{n})$

If $L_{n}$ as defined by (2.16) and $Q_{n}$ w.l.o.g. both have asymptotic
variance $1$, the Pitman efficiency of $Q_{n}$ is given by,

$\displaystyle e(Q_{n})=(\lambda_{0}(Q_{n})+\lambda_{1}(Q_{n}))^{2}$

We shall show in Theorem 2.3 that $\lambda_{1}(\widehat{C}_{n})=0$.

Proposition 2.3 follows since its assumption is that $\dot{l}_{00}=0$ and
hence $\lambda_{0}(\widehat{C}_{n})=0$.

Going further, write,

$\displaystyle\dot{l}_{00}(U,V)=a(U)+b(V)$

with $E_{0}a(U)=E_{0}b(V)=0$.

Suppose,

$\displaystyle h_{0}(x,y)=f_{0}(x)g_{0}(y).$

Define a new model $\widetilde{\mathcal{M}}$ by,

$\displaystyle\widetilde{\mathcal{M}}=\\{\widetilde{h}_{\theta}(x,y):|\theta|<1\\}$

and

$\displaystyle\widetilde{h}_{0}(x,y)=h_{0}(x,y).$

Define,

$\displaystyle\widetilde{h}_{\theta}(x,y)=\frac{h_{0}(x,y)e^{\theta(a(x)+b(y))}}{Z(\theta)}.$

where $Z(0)=1$ and $Z(\theta)$ is a normalizing function.

Then, $X$ and $Y$, or equivalently, $U$ and $V$ are independent for all
$\theta$.

Let,

$\displaystyle\Delta_{n}(\theta)\equiv\int...\int|\Pi_{i=1}^{n}h_{\theta}(x_{i},y_{i})-\Pi_{i=1}^{n}\widetilde{h}_{\theta}(x_{i},y_{i})|\Pi_{i=1}^{n}dx_{i}dy_{i}$
(2.34)

We will, show under some regularity conditions,

$\displaystyle sup\\{\Delta_{n}(\theta):|\theta|\leq\frac{M}{\sqrt{n}}\\}\to
0$ (2.35)

We prove the following in the appendix.

######  Proposition 2.4.

Let $h_{\theta},\widetilde{h}_{\theta},a,b$ be as given.

Assume, in addition, that if $l_{\theta}\equiv logh_{\theta}$,
$\big{|}\frac{\partial^{2}}{\partial\theta^{2}}l_{\theta}(x,y)\big{|}\leq
K<\infty$ for all $|\theta|\leq\epsilon$, some $\epsilon>0,K<\infty$,

Then, (2.22) holds.

Remark:  Using the techniques of Le Cam, the conditions may be relaxed to,

1) $h_{\theta}$ QMD with $\dot{l}_{\theta}(X,Y)=a(X)+b(Y)$

2) $\int
exp[\theta(a(x)+b(y))]h_{\theta}(x,y)dxdy<\infty,|\theta|\leq\epsilon,\epsilon>0$

The techniques this relaxation requires may obscure the classical argument.

This means that the variational distance between $P_{\theta,n}$, the joint
distribution of $(X_{i},Y_{i}),i=1,...,n$ under $\mathcal{M}$, and
$\widetilde{P}_{\theta,n}$, the joint distribution of the
$(X_{i},Y_{i}),i=1,...,n$ under $\widetilde{\mathcal{M}}$, goes to $0$
uniformly for $|\theta|\leq\frac{M}{\sqrt{n}}$, all $M<\infty$.

This implies that,

$\displaystyle P_{\theta_{n}n}[\sqrt{n}\widehat{C}_{n}\geq
t_{n}]=\widetilde{P}_{\theta_{n}n}[\sqrt{n}\widehat{C}_{n}\geq t_{n}]+o(1)$

for any sequence ${t_{n}}$, and $\theta_{n}=\frac{M}{\sqrt{n}}$.

Statistically, we can interpret this as follows. If $\widehat{C}_{n}$ yields
significant evidence against $H$ under $\mathcal{M}$, and
$\widetilde{\mathcal{M}}$ holds, we are being misled since, under
$\widetilde{\mathcal{M}}$, $X$ and $Y$ are always independent. But if we are
convinced that $\mathcal{M}$ holds, we would not be using ranks.

In Theorem 2.2, we shall show that this conclusion holds generally for rank
statistics which like $\widehat{C}_{n}$ are getting at the ”local dependence”
of $X$ and $Y$.

These remarks can be viewed in another way in the context of semiparametric
copula models.

####  2.3.4 Semiparametric Copula models

We start with a parametric model of densities
$\mathcal{M}=\\{h_{\theta}:|\theta|<1\\}$, where
$h_{\theta}:(0,1)^{2}\rightarrow[0,\infty)$ for all $|\theta|<1$, and
$h_{0}(x,y)\equiv 1$. Let

$\mathcal{F}:=\\{a:[0,1]\rightarrow[0,1]\text{ absolutely
continuous},a^{\prime}>0,a(0)=0,a(1)=1\\}.$

i.e., $\mathcal{F}$ is the set of all absolutely continuous, strictly
increasing transformations which map $0$ to $0$ and $1$ to $1$. Given
$q,r\in\mathcal{F}$, we may define

$h_{\theta}(x,y,q,r):=h_{\theta}(q(x),r(y))q^{\prime}(x)r^{\prime}(y),~{}~{}(x,y)\in(0,1)^{2}.$

Note that if $h_{\theta}$ is the density of $(X,Y)$, then
$h_{\theta}(\cdot,\cdot,q,r)$ is the density of the pair
$(q^{-1}(X),r^{-1}(Y))$. Define a semiparametric model by

$\displaystyle\mathcal{P}:=\\{h_{\theta}(x,y,q,r):|\theta|<1,q,r\in\mathcal{F}\\}.$

Note that we can, without loss of generality, take the domain of $h_{\theta}$
to be $R^{2}$ rather than $[0,1]^{2}$ since any statistical points we shall
make are unchanged if we consider as basic alternatives $(F_{1}(X),F_{2}(Y))$
where $F_{1},F_{2}$ are the cdf of $X,Y$ respectively, under $H$, rather than
general $X,Y$. As in the past we shall call this vector, $(U,V)$.

Note also that we could similarly consider maps $a(.)$ belonging to
$\mathcal{F}$ as having range $R$ rather than $(0,1)$. These models are
natural candidates for using ranks in relation to any inference for $\theta$
since the ”scales” a and b are unknown.

Note that $h_{0}(x,y,q,r)=q^{\prime}(x)r^{\prime}(y)$ corresponds to
independence for all $(q,r)$ pairs.

Let for $F_{\theta},G_{\theta}$ the cdf of $X,Y$ respectively,

$\displaystyle\mathcal{Q}:=\\{Q_{\theta}:(F_{\theta}(X),G_{\theta}(Y))\sim
Q_{\theta},~{}(X,Y)\sim h_{\theta},|\theta|<1\\}.$ (2.36)

Then $\mathcal{Q}$ is a parametric model which generates the same
semiparametric copula model as $\\{h_{\theta}(x,y):|\theta|<1\\}$.

A parametric copula model is one where, for every $\theta$, the marginal
distributions of $(X,Y)$ are $\mathcal{U}(0,1)$.

All classical parametric copula models, Archimedean, Gaussian, etc. are
describable as in (2.23) with, in general, multivariate $\theta$.

As usual in semiparametric theory we now carry out calculations for parametric
submodels of $\mathcal{P}$ where $q$ and $r$ depend on $\theta$ as well as
$h_{\theta}(x,y)$. We will argue that $\mathcal{Q}$ is a least favorable
submodel for estimating $\theta$ in $\mathcal{P}$.

Consider a parametric submodel, with
$\bar{\theta}=(\theta_{1},\theta_{2},\theta_{3})$, and

$\phi_{\bar{\theta}}(x,y):=h_{\theta_{1}}(x,y,q_{\theta_{2}},r_{\theta_{3}}),~{}~{}|\bar{\theta}|_{\infty}<1.$

Let $\ell_{\bar{\theta}}:=\log\phi_{\bar{\theta}}$, and let

$\nabla\ell_{\bar{0}}(x,y):=(D_{1}(x,y),D_{2}(x,y),D_{3}(x,y))$

be the gradient of $\ell$ (in the $\bar{\theta}$ variable) at
$\bar{\theta}=0$. We suppose that the family
$\\{\phi_{\bar{\theta}}:|\bar{\theta}|_{\infty}<1\\}$ is quadratic mean
differentiable at $\bar{\theta}=\bar{0}$. That is, it satisfies

$\mathbb{E}_{\bar{0}}\big{(}\phi_{\bar{\theta}}(X,Y)^{1/2}-\phi_{\bar{0}}(X,Y)^{1/2}-(1/2)\phi_{\bar{0}}(X,Y)^{1/2}(\nabla\ell_{\bar{0}}(X,Y),\bar{\theta})\big{)}^{2}=o(|\bar{\theta}|_{2}^{2}),$
(QMD)

where $\mathbb{E}_{\bar{0}}$ signifies that the random variables $(X,Y)$ are
distributed according to
$\phi_{\bar{0}}(x,y)=q_{0}^{\prime}(x)r_{0}^{\prime}(y)$. Note that

$D_{1}(X,Y)=\frac{\partial}{\partial\theta_{1}}\log
h_{\theta_{1}}(q_{\theta_{2}}(X),r_{\theta_{3}}(Y))\bigg{|}_{\bar{\theta}=0}=\dot{\ell}_{0}(q_{0}(X),r_{0}(Y)),$

(here $\dot{\ell}_{0}:=\partial\log h_{\theta}/\partial\theta|_{\theta=0}$)
and since $h_{\bar{0}}\equiv 1$, we have

$D_{2}(X,Y)=\frac{\partial}{\partial\theta_{2}}\log
q_{\theta_{2}}^{\prime}(X)\bigg{|}_{\theta_{2}=0},~{}~{}~{}D_{3}(X,Y)=\frac{\partial}{\partial\theta_{3}}\log
r_{\theta_{3}}^{\prime}(Y)\bigg{|}_{\theta_{3}=0}.$

$\mathbb{Remarks:}$ 1) Let

$\displaystyle q_{\theta}^{\prime}(x)$
$\displaystyle:=q_{0}^{\prime}(x)\big{(}1+\theta a(q_{0}(x))\big{)},$ (2.37)
$\displaystyle r_{\theta}^{\prime}(y)$
$\displaystyle:=r_{0}^{\prime}(y)\big{(}1+\theta b(r_{0}(y))\big{)},$

where $|a|<1,|b|<1$, and $\mathbb{E}a(U)=\mathbb{E}b(U)=0$, for
$U\sim\mathrm{Unif}(0,1)$. If $\\{h_{\theta}:|\theta|<1\\}$ satisfies
assumptions I and II, then the parametric copula model defined by (2.23) above
satisfies (QMD).

The tangent spaces (using the notation of Bickel et al. (1998)) of the model
defined by (2.37) at $\theta_{1}=\theta_{2}=\theta_{3}=0$ are

$\dot{\mathcal{P}}_{\theta_{1}}=[\dot{\ell}_{0}(q_{0}(X),r_{0}(Y))],~{}~{}\dot{\mathcal{P}}_{\theta_{2}}=[a(q_{0}(X))],~{}~{}\dot{\mathcal{P}}_{\theta_{3}}=[b(r_{0}(Y))].$

Since the set of $a,b$ as above are dense in
$L_{2}^{0}([0,1]):=\\{c(U):\mathbb{E}[c(U)^{2}]<\infty,\mathbb{E}c(U)=0\\}$,
the tangent spaces of the semiparametric copula model are

$\displaystyle\dot{\mathcal{P}}_{\theta}=[\dot{\ell}_{0}(q_{0}(X),r_{0}(Y))],$
$\displaystyle\dot{\mathcal{P}}_{q}=\\{f(q_{0}(X)):f\in
L_{2}^{0}([0,1])\\},~{}~{}\dot{\mathcal{P}}_{r}=\\{f(r_{0}(Y)):f\in
L_{2}^{0}([0,1])\\}.$

While $\dot{\ell}_{0}(q_{0}(X),r_{0}(Y))$ is the score function for the base
model $\\{h_{\theta}(\cdot,\cdot,q_{0},r_{0}):|\theta|<1\\}$, we now obtain
that

$\displaystyle\dot{\ell}_{0}^{*}(X,Y):=\dot{\ell}_{0}(q_{0}(X),r_{0}(Y))-\mathbb{E}[\dot{\ell}_{0}(q_{0}(X),r_{0}(Y))~{}|~{}X]-\mathbb{E}[\dot{\ell}_{0}(q_{0}(X),r_{0}(Y))~{}|~{}Y]$
(2.38)

is the efficient score function for the model $\mathcal{P}$, since
$\mathbb{E}[\dot{\ell}_{0}(q_{0}(X),r_{0}(Y))~{}|~{}X]$ is the projection of
$\dot{\ell}_{0}(q_{0}(X),r_{0}(Y))$ on $\dot{\mathcal{P}}_{q}$ and similarly
for $\dot{\mathcal{P}}_{r}$ (see (Bickel et al., 1998, Chapter 3)).

Consider the submodel of $\mathcal{P}$, given by (2.37), with
$\theta_{1}=\theta_{2}=\theta_{3}=\theta$, and

$a(q_{0}(X))=-\mathbb{E}[\dot{\ell}_{0}(q_{0}(X),r_{0}(Y))~{}|~{}X],~{}~{}b(r_{0}(Y))=-\mathbb{E}[\dot{\ell}_{0}(q_{0}(X),r_{0}(Y))~{}|~{}Y].$

Given assumptions I and II, we have that (QMD) holds, and hence this model is
also quadratic mean differentiable at $\theta=0$, with score function
$\dot{\ell}_{0}^{*}$. For estimation, this means that if an estimate is
regular and linear on $\mathcal{P}$, then its asymptotic variance
$\sigma^{2}/n\geq(I^{*})^{-1}/n$, where

$I^{*}:=\mathbb{E}_{0}[\dot{\ell}_{0}^{*}(q_{0}(X),r_{0}(Y))^{2}].$

For a rank test statistic $T_{n}$ such that under $\theta=0$, we have that

$\displaystyle\bigg{(}\sqrt{n}T_{n},\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\dot{\ell}_{0}^{*}(X_{i},Y_{i})\bigg{)}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}N\bigg{(}\begin{pmatrix}0\\\
0\end{pmatrix},\begin{pmatrix}\sigma^{2}&\rho^{*}\sigma\sqrt{I^{*}}\\\
\rho^{*}\sigma\sqrt{I^{*}}&I^{*}\end{pmatrix}\bigg{)},$

Then as in (2.17), the Pitman efficiency of $T_{n}$ with respect to the
optimal asymptotic test is

$\displaystyle e^{*}(T_{n})=(\rho^{*})^{2}.$

2) We can exhibit explicit families $\\{q_{\theta}:|\theta|<1\\}$,
$\\{r_{\theta}:|\theta|<1\\}$ such that the parametric model
$\\{h_{\theta}(\cdot,\cdot,q_{\theta},r_{\theta}):|\theta|<1\\}$ has score
function $\dot{\ell}_{0}^{*}$. Set $q_{\theta}:=F_{\theta}^{-1}$,
$r_{\theta}:=G_{\theta}^{-1}$, where $F_{\theta},G_{\theta}$ are the
respective marginal cdfs of $X,Y$ under $h_{\theta}$. (Note then the
distribution $h_{\theta}(\cdot,\cdot,q_{\theta},r_{\theta})$ is given by the
law of $(F_{\theta}(X),G_{\theta}(Y))$, if $(X,Y)$ is distributed according to
$h_{\theta}$. So we are essentially going from the semiparametric copula
generated model back to the parametric copula model.) Let $\dot{s}_{0}$ denote
the score function at $\theta=0$ for this model. A change of variables
calculation then gives that

$\dot{s}_{0}(X,Y)=\dot{\ell}_{0}(q_{0}(X),r_{0}(Y))-\frac{1}{f_{0}(X)}\frac{\partial
f_{\theta}}{\partial\theta}(q_{0}(X))\bigg{|}_{\theta=0}-\frac{1}{g_{0}(Y)}\frac{\partial
g_{\theta}}{\partial\theta}(r_{0}(Y))\bigg{|}_{\theta=0}.$

From semi parametric theory it follows that the efficient score function at
$\theta=0$ is the projection of $\dot{\ell}_{0}$ on the ortho complement of
the nuisance tangent space $\dot{\mathcal{P}}_{q}+\dot{\mathcal{P}}_{r}$.
Without loss of generality, we can take $q_{0}$ and $r_{0}$ the identity. Thus
$\dot{\mathcal{P}}_{q}+\dot{\mathcal{P}}_{r}=\Lambda_{1}$ as defined in (2.13)
for n=1.

####  2.3.5  Local power behaviour of classical statistics

The properties of $\widehat{T}_{BKR}$, $\widehat{T}_{S}$ are essentially well
known.

It is obvious, for instance, from the expansion

$\displaystyle\widehat{T}_{S}=\frac{12}{n}\sum_{i=1}^{n}{(U_{i}-\frac{1}{2})(V_{i}-\frac{1}{2})}+o_{p}(n^{-\frac{1}{2}}).$
(2.39)

that $\widehat{T}_{S}$ has positive Pitman efficiency in any $\mathcal{M}$,
unless $\dot{l}_{0}(X,Y)\perp(U-\frac{1}{2})(V-\frac{1}{2})$.

Nevertheless, the following theorem may not be known in full generality.

######  Theorem 2.1.

If $\dot{l}_{0}(X,Y)\neq a(X)+b(Y)$ for some $a,b$, then $\widehat{T}_{BKR}$
is asymptotically unbiased.

As we have argued in the previous section, given a semiparametric formulation
of a model, a score function in $\Lambda_{1}$, corresponds to models in which
dependence cannot be measured reliably.

$\mathbb{Proof:}$ By general theory, under $H$,

$\displaystyle n\widehat{T}_{BKR}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\int[T^{*}]^{2}(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}})d\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}}.$

and, under $\\{P_{n\theta_{n}}\\}$, if
$(T_{n}(.),\frac{1}{\sqrt{n}}\sum_{i=1}^{n}l_{0}^{*}(X_{i},Y_{i}))\stackrel{{\scriptstyle
d}}{{\longrightarrow}}(T^{*}(.),Z)$,

$\displaystyle n\widehat{T}_{BKR}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\int(T^{*}(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}})+tC(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}}))^{2}d\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}}.$

where

$\displaystyle
C(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}})=cov(T^{*}(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}}),Z)$

By classical theory, we can represent, for $Z_{ij}$ iid $N(0,1)$,

$\displaystyle\int[T^{*}]^{2}(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}})d\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}}$
$\displaystyle=\sum_{i,j\geq 1}\lambda_{ij}Z_{ij}^{2}$

where

$\displaystyle\sqrt{\lambda_{ij}}Z_{ij}=\int
T^{*}(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}})f_{ij}(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}})d\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}}$
(2.46)

and,

$\displaystyle
f_{ij}(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}})=2sin(\pi
iu)sin(\pi jv).$

Then,

$\displaystyle\int(T^{*}(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}})+tC(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}}))^{2}d\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}}$
$\displaystyle\sim\sum_{i,j\geq 1}\lambda_{ij}(Z_{ij}+tC_{ij})^{2}.$

Here, as shown by Blum et al (1961) Blum et al. (1961),
$\lambda_{ij}=\frac{1}{\pi^{4}i^{2}j^{2}}$,

$\displaystyle C_{ij}=\int
C(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}})f_{ij}(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}})d\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}}$

and $f_{ij}$ are the eigenfunctions of the covariance kernel given in (2.14).
By Lemma A1, $C(.)\not\equiv 0$. By Zygmund Zygmund (2002) Problems 1.8.4 and
1.8.6, the $f_{ij}$ are a complete orthonormal system in $L_{2}[0,1]^{2}$.
Therefore, if $C(.)\not\equiv 0$, some $C_{ij}\neq 0$. Then, if $t\neq 0$, the
distribution of $(Z_{ij}+tC_{ij})^{2}$ is stochastically larger than that of
$Z_{ij}^{2}$ and the theorem follows. ∎

As, is well known in the context of goodness of fit tests Lehmann and Romano
(2006), no test sequence can have substantial local power in more than a
finite set of directions, essentially choices of $\dot{l}_{0}$ in our context.
This carries over to semiparametric hypotheses within nonparametric models,
such as independence generally, see for instance Bickel, Ritov and Stoker
(2006)Bickel et al. (2006). This is clear from consideration of
$\widehat{T}_{BKR}$, where it is natural to take as primitive directions, the
elements of the Fourier basis, that is,
$\dot{l}_{0}(F_{1}^{-1}(U),F_{2}^{-1}(V))=f_{ij}(U,V)$. The weight
$\frac{1}{\pi^{4}i^{2}j^{2}}$ placed on each of these in the limit
distribution of $\widehat{T}_{BKR}$ diminishes as a function of $i,j$,
corresponding to their ”wiggliness”. Bickel, Ritov and Stoker (2006) Bickel et
al. (2006) discuss various ways of favoring preferred directions while
retaining some local power in all directions. We will turn to this topic in
the contex of the $f_{ij}$ and other orthonormal bases elsewhere.

In the remainder of this section, we will

1) Make explicit the construction of rank statistics corresponding to the same
statistics in terms of the $(U_{i},V_{i})$ (which are not observed)

2)Indicate how such statistics can be used for a simpler purpose,
approximating $\widehat{T}_{BKR}$ to arbitrary precision in $O(n)$ operations,
as well as building blocks for selective test of independence.

We use the following lemma, proved in the Appendix.

######  Lemma 2.2.

Let $a,b\in L_{2}[0,1]$. Let $A(u)\equiv\int_{0}^{u}a(s)ds$,
$B(v)\equiv\int_{0}^{v}b(t)dt$. Let $(R_{i},S_{i}),i=1,...,n$ be the ranks of
the $(U_{i},V_{i})$ coordinatewise where
$U_{i}=F_{1}(X_{i}),V_{i}=F_{2}(Y_{i})$. Let $\bar{A}(u)=A(u)-EA(u)$,
$\bar{B}(v)=B(v)-EB(v)$. Then, under $H$,

$\displaystyle\int_{0}^{1}\int_{0}^{1}T_{n}(u,v)a(u)b(v)dudv=\frac{1}{\sqrt{n}}\sum_{k=1}^{n}{\bar{A}(\frac{R_{k}}{n})\bar{B}(\frac{S_{k}}{n})}+o_{p}(n^{-\frac{1}{2}})$
(2.47)

$\mathbb{Remark:}$ If $a$ is bounded,
$EA(U)=\frac{1}{n}\sum_{k=1}^{n}{A(\frac{k}{n})}+O(\frac{1}{n})$

Now apply Lemma 2.2 to the $f_{ij}$ and obtain,

$\displaystyle\widehat{T}_{ij}\equiv\int
T_{n}(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}})f_{ij}(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}})d\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}}=\frac{2}{\sqrt{n}}\sum_{k=1}^{n}\bigg{[}\frac{cos(\frac{\pi
iR_{k}}{n})cos(\frac{\pi jS_{k}}{n})}{(\pi
ij)^{2}}\bigg{]}+o_{p}(n^{-\frac{1}{2}})$ (2.54)

since $\int_{0}^{u}sin(\pi it)dt=\frac{1-cos(\pi iu)}{\pi i}$ and
$\int_{0}^{1}cos(\pi iu)du=0$.

By Proposition 2.1, $\\{\widehat{T}_{ij}:1\leq i,j\leq N\\}$ are
asymptotically jointly Gaussian with mean 0, independent, and with variance
$\\{[\pi^{4}i^{2}j^{2}]^{-1}:i,j\leq N\\}$. The independence holds because
$f_{ij}(.)$ are an orthornormal basis of $L_{2}[0,1]^{2}$, eigenfunctions of
the covariance kernel of $T^{*}$ and (2.27).

Moreover, by standard Fourier analysis,

$\displaystyle
T_{n}(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}})=\sum_{i,j\geq
1}{\widehat{T}_{ij}f_{ij}(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}})}\textit{
a.e }$

and

$\displaystyle n\widehat{T}_{BKR}=\int
T_{n}^{2}(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}})d\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}}=\sum_{i,j\geq
1}\widehat{T}_{ij}^{2}$ (2.59)

If

$\displaystyle\widehat{T}_{n,M}^{2}(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}})\equiv\sum_{i,j=1}^{M}\widehat{T}_{ij}f_{ij}(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}})$
(2.64)

then,

$\displaystyle\int
T_{n,M}^{2}(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}})d\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}}=\sum_{i,j=1}^{M}\widehat{T}_{ij}^{2}$

and

$\displaystyle\int(T_{n,M}(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}})-T_{n}(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}}))^{2}d\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}}\to
0\textit{ in probability as }M\to\infty$

We have proved,

$\mathbb{Proposition2.5:}$ Under contiguous alternatives, $n\widehat{T}_{BKR}$
can be approximated arbitrarily closely by $\sum_{i,j\geq
1}^{M}\widehat{T}_{ij}^{2}$ and these statistics can be computed in $O(n)$
steps.

In fact we can use the $\widehat{T}_{ij}$ for selective inference since they
are symptotically independent. We can use them in lexicographic order in the
$(i,j)$ basis, corresponding to their wiggliness. Note again that we are using
the fact that $sin(\pi iu)sin(\pi jv)$ are both eigenfunctions of the
covariance kernel of $T_{n}(.,.)$ and an orthonormal basis of
$L_{2}[0,1]^{2}$.

It is attractive to consider alternative orthonormal basis which localize
failure of independence more specifically. A natural candidate based on
Rademacher functions, is given as follows. Choose a scale of interest,
$\Delta=2^{-N}$. Let

$\displaystyle\Psi_{p}(u)=\Delta\Psi(\frac{u}{\Delta}-p),\quad
p=0,...,2^{N}-1$ (2.65)

where

$\displaystyle\Psi(u)=sgn(u-\frac{1}{2})$

By construction, the $\Psi_{p}(.)$ are orthonormal. We could take $\Delta>0$
generally, but our choice enables us to vary $N=0,1,2,...$ and create a full
orthonormal basis of

$\displaystyle L_{2}^{0}(0,1)\equiv\\{h\in L_{2}(0,1):\int h(u)du=0\\}.$

Let,

$\displaystyle\widehat{S}_{N,p_{1},p_{2}}$
$\displaystyle=\int_{0}^{1}\int_{0}^{1}T_{n}(u,v)\Psi_{p_{1}}(u)\Psi_{p_{2}}(v)dudv$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\lambda_{p_{1}}(\frac{R_{i}}{n})\lambda_{p_{2}}(\frac{S_{i}}{n})+o_{p}(n^{-\frac{1}{2}})$

under $H$, where

$\displaystyle\lambda_{p}(u)$ $\displaystyle=2^{-N}\lambda_{0}(2^{N}u-p)$
$\displaystyle\lambda_{0}(u)$
$\displaystyle=\int_{u}^{1}\Psi(s)ds-\frac{1}{4}$ $\displaystyle=t1(0\leq
t\leq\frac{1}{2})+(1-t)1(\frac{1}{2}\leq t\leq 1)-\frac{1}{4}$

Under $H$, the $\widehat{S}_{N,p_{1},p_{2}}$ are asymptotically Gaussian with
mean 0, (jointly), iid for fixed N with variance $\frac{2^{-N}}{48}$, but
dependent as N varies.

Selective inference is done for successive scales. The approach can be
naturally extended to independence of more than 2 variables. An approach
related to this proposal has been put forward by Zhang Zhang (2019).

###  2.4 Local independence and a family of test statistics and parameters
related to $\widehat{C}_{n}$

Chatterjee’s $\widehat{C_{n}}$ and recent work of Lin and Han Lin and Han
(2021) and Deb, Ghosal and Sen Deb et al. (2020) suggest a family of
parameters based on a representation of the joint distribution of a pair
$(X,Y)$ with $F$ continuous which leads to novel parameters measuring
dependence and test statistics for H.

If F is the probability distributions of a pair $(X,Y)$, we can consider it in
the form, $(F_{1}(.),F_{2}(.|.))$ where $F_{2}(y|x)$ is the cdf of the regular
conditional probability distribution of $Y$ given $X$. We can extend this
representation by considering $Y^{(1)},...,Y^{(m)}$ conditionally iid with
distribution $F_{2}(.|X)$. Call the joint distribution of
$(X,Y^{(1)},...,Y^{(m)})$, $F^{(m)}$. With this notation, the Chatterjee and
Dette et al. parameter in form (1.2) if F is continuous is,

$\displaystyle T(F)=1-3E_{F^{(}2)}\\{|F_{2}(Y^{(1)})-F_{2}(Y^{(2)})|\\}$
(2.66)

This point of view leads naturally to considering the process defined on
$[0,1]^{3}$ by,

$\displaystyle\widehat{C_{L}}(u,v_{1},v_{2})\equiv\frac{1}{n-1)}\sum_{i=1}^{n-1}1(\widehat{F_{1}}(X_{(i)})\leq
u,\widehat{F_{2}}(Y_{[i]})\leq v_{1},\widehat{F_{2}}(Y_{[i+1]})\leq v_{2})$

where, if $(X_{1},Y_{1}),...,(X_{n},Y_{n})$ are the original sample,
$X_{(1)}<...<X_{(n)}$ are the ordered $X_{i}$ and
$(X_{(i)},Y_{[i]})=(X_{D_{i}},Y_{D_{i}})$ where $D_{i}=j$ iff $X_{(i)}=X_{j}$,
and $\widehat{F}_{j}$ are the usual empirical marginal cdf.

Thus, in the notation of (1.1),

$\displaystyle\widehat{C_{L}}(u,v_{1},v_{2})=\frac{1}{n-1}\sum_{i=1}^{[nu]}1(\frac{r_{i}}{n}\leq
v_{1},\frac{r_{i+1}}{n}\leq v_{2}).$

Next, define in analogy to $T_{n}(.,.)$,

$\displaystyle
S_{n}(u,v_{1},v_{2})=\sqrt{n}(\widehat{C_{L}}(u,v_{1},v_{2})-C_{L}(u,v_{1},v_{2}))$

where,

$\displaystyle
C_{L}(u,v_{1},v_{2})=\int_{0}^{u}F_{2}^{*}(v_{1}|Q_{1}(s))F_{2}^{*}(v_{2}|Q_{1}(s))ds$

and $F_{2}^{*}$ is the conditional cdf of $F_{2}(Y)$ given $X=x$.

In the case we pursue, $X,Y$ independent and $F$ continuous,
$C_{L}(u,v_{1},v_{2})=uv_{1}v_{2}$ and we focus on,

$\displaystyle
S_{n}(v_{1},v_{2})=\sqrt{n}(\widehat{C_{L}}(v_{1},v_{2})-v_{1}v_{2})$ (2.67)

where $\widehat{C_{L}}(v_{1},v_{2})\equiv\widehat{C_{L}}(1,v_{1},v_{2})$.

Rank statistics such as $\widehat{T}_{BKR},\widehat{T}_{S}$ have analogues,
(neglecting terms of $o(\frac{1}{n})$).

In particular, define

$\displaystyle\widehat{T}_{S}^{(L)}=12(\int_{0}^{1}\int_{0}^{1}v_{1}v_{2}d\widehat{C_{L}}(v_{1},v_{2})-\frac{1}{4})$
$\displaystyle=\frac{12}{n-1}\sum_{i=1}^{n-1}(\frac{r_{i}}{n}\frac{r_{i+1}}{n}-\frac{1}{4})$
(2.68)

All such statistics can evidently be computed using the $(r_{i},r_{i+1})$ just
as their global analogues use $(R_{i},S_{i})$.

$\mathbb{Remark:}$ Such processes and statistics have been defined previously
in a different context, see for example, Genest et al (2007)Genest et al.
(2007).

We shall see in Theorem 2.2 that $\widehat{T}_{S}^{(L)}$ and all statistics
based on $\widehat{C}_{L}$ have the same poor local power behaviour we claimed
for $\widehat{C}_{n}$.

The following result will prove useful. Its proof is in the appendix.

######  Lemma 2.3.

Suppose $(U_{i},V_{i}),i=1,...,n$ are i.i.d $\mathcal{U}(0,1)$. Define, as
below, $V_{[i]}=V_{j}$ iff $U_{j}=U_{(i)}$ where $U_{(1)}<...<U_{(n)}$ are the
ordered $U_{i}$. Given $g\in L_{2}[0,1]^{2}$, define
$g_{1}(s)\equiv\int_{0}^{1}g(s,t)dt$,
$g_{2}(t)\equiv\int_{0}^{1}g(s,t)ds$. Let $a,h^{(1)},h^{(2)}\in
L_{2}[0,1]^{2}$ and use the subscript notation defined, noting that
$h_{1}^{(j)}(V_{1})=E[h^{(j)}(V_{1},V_{2})|V_{1}],h_{2}^{(j)}(V_{1})=E[h^{(j)}(V_{2},V_{1})|V_{1}]$,
$j=1,2$ etc.

Let,

$\displaystyle A_{n}\equiv\frac{1}{\sqrt{n}}\sum_{k=1}^{n}a(U_{(k)},V_{[k]})$
$\displaystyle
H_{n}^{(i)}\equiv\frac{1}{\sqrt{n}}\sum_{k=1}^{n-1}h^{(i)}(V_{[k]},V_{[k+1]})$

Then, $(A_{n},H_{n}^{(1)},H_{n}^{(2)})\stackrel{{\scriptstyle
d}}{{\longrightarrow}}(Z_{0},Z_{1},Z_{2})$

where $(Z_{0},Z_{1},Z_{2})$ are jointly Gaussian, mean 0 and

$\displaystyle Var(Z_{0})=Var(a(U,V))$

$\displaystyle
Var(Z_{1})=Var(h^{(1)}(V_{1},V_{2}))+Var(h_{1}^{(1)}(V_{1}))+Var(h_{2}^{(1)}(V_{2}))$
(2.69) $\displaystyle
Cov(Z_{0},Z_{j})=Cov(a_{2}(V_{1}),h_{1}^{(j)}(V_{1})+h_{2}^{(j)}(V_{1})),\quad
j=1,2.$ (2.70) $\displaystyle
Cov(Z_{1},Z_{2})=Cov(h^{(1)}(V_{1},V_{2}),h^{(2)}(V_{1},V_{2}))+Cov(h_{1}^{(1)}(V_{1}),h_{1}^{(2)}(V_{1}))+Cov(h_{2}^{(1)}(V_{2}),h_{2}^{(2)}(V_{2}))$
(2.71)

Let

$\displaystyle
h(\hbox{\oalign{$\bm{v}$\crcr$\scriptscriptstyle\bm{\sim}$}},w_{1},w_{2})=1(w_{1}\leq
v_{1})1(w_{2}\leq v_{2})-v_{1}v_{2}$

and define, consistent with our general notation,

$\displaystyle
h_{1}(\hbox{\oalign{$\bm{v}$\crcr$\scriptscriptstyle\bm{\sim}$}},w_{1})=\int_{0}^{1}h(\hbox{\oalign{$\bm{v}$\crcr$\scriptscriptstyle\bm{\sim}$}},w_{1},w_{2})dw_{2}=(1(w_{1}\leq
v_{1})-v_{1})v_{2}$

and $h_{2}(\hbox{\oalign{$\bm{v}$\crcr$\scriptscriptstyle\bm{\sim}$}},w_{2})$
analogously,

Then,

$\displaystyle
S_{n}(v_{1},v_{2})=\frac{1}{\sqrt{n}}\sum_{k=1}^{n-1}h(\hbox{\oalign{$\bm{v}$\crcr$\scriptscriptstyle\bm{\sim}$}},\frac{r_{k}}{n},\frac{r_{k+1}}{n})$

Let

$\displaystyle\widetilde{h}(\hbox{\oalign{$\bm{v}$\crcr$\scriptscriptstyle\bm{\sim}$}},w_{1},w_{2})=h(\hbox{\oalign{$\bm{v}$\crcr$\scriptscriptstyle\bm{\sim}$}},w_{1},w_{2})-h_{1}(\hbox{\oalign{$\bm{v}$\crcr$\scriptscriptstyle\bm{\sim}$}},w_{1})-h_{2}(\hbox{\oalign{$\bm{v}$\crcr$\scriptscriptstyle\bm{\sim}$}},w_{2})$
(2.80)

Define,

$\displaystyle
R_{n}(\hbox{\oalign{$\bm{v}$\crcr$\scriptscriptstyle\bm{\sim}$}})$
$\displaystyle\equiv\frac{1}{\sqrt{n}}\sum_{i=1}^{n-1}h(\hbox{\oalign{$\bm{v}$\crcr$\scriptscriptstyle\bm{\sim}$}},V_{[i]},V_{[i+1]})$
$\displaystyle
Q_{n}(\hbox{\oalign{$\bm{v}$\crcr$\scriptscriptstyle\bm{\sim}$}})$
$\displaystyle\equiv\frac{1}{\sqrt{n}}\sum_{i=1}^{n-1}\widetilde{h}(\hbox{\oalign{$\bm{v}$\crcr$\scriptscriptstyle\bm{\sim}$}},V_{[i]},V_{[i+1]})$
$\displaystyle\Lambda_{n}^{*}(\hbox{\oalign{$\bm{v}$\crcr$\scriptscriptstyle\bm{\sim}$}})$
$\displaystyle\equiv\frac{1}{\sqrt{n}}\sum_{i=1}^{n-1}(h(\hbox{\oalign{$\bm{v}$\crcr$\scriptscriptstyle\bm{\sim}$}},V_{[i]},V_{[i+1]})-\widetilde{h}(\hbox{\oalign{$\bm{v}$\crcr$\scriptscriptstyle\bm{\sim}$}},V_{[i]},V_{[i+1]}))$
$\displaystyle=\frac{1}{\sqrt{n}}\sum_{i=1}^{n-1}(h_{1}(\hbox{\oalign{$\bm{v}$\crcr$\scriptscriptstyle\bm{\sim}$}},V_{[i]}))+h_{2}(\hbox{\oalign{$\bm{v}$\crcr$\scriptscriptstyle\bm{\sim}$}},V_{[i+1]}))$

Also, by definition,

$\displaystyle
Q_{n}(\hbox{\oalign{$\bm{v}$\crcr$\scriptscriptstyle\bm{\sim}$}})=R_{n}(\hbox{\oalign{$\bm{v}$\crcr$\scriptscriptstyle\bm{\sim}$}})-\Lambda_{n}^{*}(\hbox{\oalign{$\bm{v}$\crcr$\scriptscriptstyle\bm{\sim}$}}).$

Here is an analogue to Proposition 2.1.

######  Theorem 2.2.

If $X$ and $Y$ are independent and $F_{1},F_{2}$ are continuous, then,

  1. (i)

$S_{n}(v_{1},v_{2})=Q_{n}(v_{1},v_{2})+o_{p}(1)$

  2. (ii)

$\Lambda_{n}^{*}\stackrel{{\scriptstyle d}}{{\longrightarrow}}\Lambda^{*}$
and

  3. (iii)

$S_{n}(.,.)\stackrel{{\scriptstyle d}}{{\longrightarrow}}S(.,.)$

where $S(.,.)$ is a Gaussian field with mean 0 and

if
$\hbox{\oalign{$\bm{a}$\crcr$\scriptscriptstyle\bm{\sim}$}}=(a_{1},a_{2}),\hbox{\oalign{$\bm{b}$\crcr$\scriptscriptstyle\bm{\sim}$}}=(b_{1},b_{2})$,
then

$\displaystyle
Cov(S(\hbox{\oalign{$\bm{a}$\crcr$\scriptscriptstyle\bm{\sim}$}}),S(\hbox{\oalign{$\bm{b}$\crcr$\scriptscriptstyle\bm{\sim}$}}))=Cov(T^{*}(\hbox{\oalign{$\bm{a}$\crcr$\scriptscriptstyle\bm{\sim}$}}),T^{*}(\hbox{\oalign{$\bm{b}$\crcr$\scriptscriptstyle\bm{\sim}$}}))+Cov(\Lambda^{*}(\hbox{\oalign{$\bm{a}$\crcr$\scriptscriptstyle\bm{\sim}$}}),\Lambda^{*}(\hbox{\oalign{$\bm{b}$\crcr$\scriptscriptstyle\bm{\sim}$}}))+o(1)$
(2.93)

where $T^{*}$ is given in (2.14), and $\Lambda^{*}(v_{1},v_{2})\sim
v_{1}E_{2}(v_{2})+v_{2}E_{2}(v_{1})$.

$\mathbb{Remark:}$ We can interpret (2.35) as,

$\displaystyle
S(\hbox{\oalign{$\bm{a}$\crcr$\scriptscriptstyle\bm{\sim}$}})\sim
T^{*}(\hbox{\oalign{$\bm{a}$\crcr$\scriptscriptstyle\bm{\sim}$}})+\Lambda^{*}(\hbox{\oalign{$\bm{a}$\crcr$\scriptscriptstyle\bm{\sim}$}})$

where $T^{*}$ and $\Lambda$, are $\mathbb{independent}$ mean 0 Gaussian
processes and $T^{*}(.)$ has the covariance given in (2.14). Here ”$\sim$”
indicate distributional equivalence since the expression arises from the CLT
for 2 dependent variables Orey(1958) Orey (1958).

$\mathbb{Proof:}$ As with Theorem 2.1, we need to show tightness and joint
FIDI convergence of the 2 processes,$S_{n}-\Lambda^{*}_{n}$ and
$\Lambda^{*}_{n}$, and prove they have continuous sample surfaces. The delta
method is applied as in Theorem 2.1.

Given $\\{U_{(1)}<...<U_{(n)}\\}\equiv\mathcal{B}_{n}$, the $V_{[i]}$ are iid
$\mathcal{U}(0,1)$ and we can just write $V_{i}$.

Apply the 2 dependent CLT (Orey(1958)Orey (1958)) to obtain that, since
$E[Q_{n}(.)|\mathcal{B}_{n}]=E[\Lambda_{n}^{*}(.)|\mathcal{B}_{n}]=0$,

$\displaystyle(Q_{n}(.),\Lambda_{n}^{*}(.))\to(Q,\Lambda^{*})$

where $Q$ and $\Lambda^{*}$ are independent, and the covariances are as
specified by applying Lemma 2.1.

$\\{\Lambda_{n}^{*}\\}$ is evidently tight. $Q_{n}(.)$ is also by a direct
calculation showing that

$\displaystyle
E(\Delta_{\epsilon}Q_{n})^{2}(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}})(\Delta_{\epsilon}Q_{n})^{2}(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}}+\hbox{\oalign{$\bm{\epsilon}$\crcr$\scriptscriptstyle\bm{\sim}$}})\leq
C\epsilon^{2}$ (2.100)

where
$\hbox{\oalign{$\bm{\epsilon}$\crcr$\scriptscriptstyle\bm{\sim}$}}=(\epsilon_{1},\epsilon_{2})$
and

$\displaystyle\Delta_{\epsilon}G(u_{1},u_{2})\equiv
G(u_{1}+\epsilon_{1},u_{2}+\epsilon_{2})-G(u_{1}+\epsilon_{1},u_{2})-G(u_{1},u_{2}+\epsilon_{2})+G(u_{1},u_{2})\qquad\qed$

We generalize Proposition 2.3.

######  Theorem 2.3.

Suppose $H$ holds and the condition of Proposition 2.3 is satisfied.

Then, $S_{n}(.)$ is asymptotically independent of
$L_{n}\equiv\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\dot{l}_{0}(X_{i},Y_{i})$. Thus,
any test statistic based on $S_{n}(.)$ which has a joint limit with $L_{n}$
has no power.

Proof:  In view of Theorem 2.2, we need only check that,

$\displaystyle\lim_{n}cov_{0}\bigg{(}\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\big{(}h_{1}(\hbox{\oalign{$\bm{v}$\crcr$\scriptscriptstyle\bm{\sim}$}},V_{[i]})+h_{2}(\hbox{\oalign{$\bm{v}$\crcr$\scriptscriptstyle\bm{\sim}$}},V_{[i+1]})\big{)},L_{n}\bigg{)}=0$
(2.105)

and

$\displaystyle\lim_{n}cov_{0}\bigg{(}\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\widetilde{h}(\hbox{\oalign{$\bm{v}$\crcr$\scriptscriptstyle\bm{\sim}$}},V_{[i]},V_{[i+1]}),L_{n}\bigg{)}=0$
(2.108)

since joint Gaussianity of these statistics follows from Orey’s theorem once
we write (with $X=U,Y=V$)

$\displaystyle
L_{n}=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\dot{l}_{0}(U_{[i]},V_{[i]})$

(2.42) follows by substituting $\dot{l}_{0}$ for $a(.,.)$ and $h_{j}$ for
$h^{(j)}$ in Lemma 2.3.

By assumption, $a_{2}=0$. Similarly, by substituting $\widetilde{h}$ by
$h_{j}$, (2.43) follows.∎

###  2.5 Discussion

1) If F is not continuous, we can have $(X_{1},Y_{1}),(X_{2},Y_{2})$ such that
$X_{1}=X_{2}$ or $Y_{1}=Y_{2}$ or both. In those cases, the distribution of
global rank statistics under H depends on the underlying $F_{1},F_{2}$. For
local statistics, as Chatterjee points out, ties occurring only in X can be
broken randomly.

Valid tests can be carried out by conditioning on $\widehat{F}_{1}(.)$ and
$\widehat{F}_{2}(.)$. For instance, if $(X,Y)$ is discrete, we would be as
usual conditioning on the row sums of a contingency table.

2) Even in the continuous case, tests based on statistics such as
$\widehat{T}_{BKR}$ or $\widehat{T}_{BKR}^{(L)}$are naturally carried out
using Monte Carlo approximations to the null distributions using simulations
of the $(U_{i},V_{i})$ under $H$.

##  3 Functional Dependence

In this section we will discuss property $C)$, functional dependence of the
Chatterjee-Dette et al. parameter and a principle leading to $R,C$, and a
number of other parameters defining functional dependence in different
contexts. We change notation slightly here letting $F_{X}$ and $G_{Y}$ denote
the marginal cdf of $X$ and $Y$.

###  3.1 A Measure of functional dependence

We begin with the solution to a simple problem. For $f_{0}(Y)\in L^{2}(Y)$,
let

$\displaystyle M(X,f_{0}(Y)):=sup\\{\rho(f_{0}(Y),g(X))^{2},:g(X)\in
L^{2}(X)\\},$

Here $\rho$ is the correlation coefficient.

By the correlation inequality,

$\displaystyle M(X,f_{0}(Y))=\frac{Var(E(f_{0}(Y)|X))}{Var(f_{0}(Y))}.$ (3.1)

This measure has the property that $M(X,f_{0}(Y))=1$ iff $f_{0}(Y)=h(X)$ a.s.
for some $h$.

###  3.2 C in this context

The Chatterjee-Dette et al parameter emerges from this point of view as
follows.

Note that for continuous $F_{X},G_{Y}$, $Y=h(X)$ iff
$G_{Y}(Y)=\widetilde{h}(F_{X}(X))$ a.e. where

$\displaystyle\widetilde{h}(x)=G_{Y}^{-1}\widetilde{h}F_{X}(x)$

and $F^{-1}(u)\equiv inf\\{x:F(x)\geq u\\}$.

We obtain a measure of functional dependence of $Y$ on $X$ in (3.1) by taking
$f_{0}(y)\equiv y$. Then,

$\displaystyle M(X,Y)$ $\displaystyle=\frac{Var(E(Y|X))}{Var(Y)}$ (3.2)
$\displaystyle=1-\frac{E(Var(Y|X))}{Var(Y)}.$

This satisfies the functional dependence requirements C) but not B).
Chatterjee and Dette et al. implicitly use that for $Y\in R$, independence is
equivalent to

$\displaystyle P(Y\leq y|X)=P(Y\leq y)\textit{ a.s.}$ (3.3)

which corresponds to ,

$\displaystyle M(X,1(Y\leq y))=\frac{E(Var(1(Y\leq y)|X))}{Var(1(Y\leq y))}=1$
(3.4)

for almost all $y$.

Chatterjee then achieved the goal of of characterizing independence by
integrating the numerator and denominator in (3.4) separately in y and
defining C as in (1.2).

###  3.3 Extensions

A natural extension of (3.1) is to specify parameters having the extremal
property for $g$ belonging to subsets of $L_{2}(X)$, and also for $Y,X$ with
values in $R^{p}$ or more general spaces. A natural extension of this type is
to define for $Y\in R^{p}$,

$\displaystyle M_{L}(X,Y)\equiv max\\{M(X,f(Y)),f\in\mathcal{F}\\}$ (3.5)

where $\mathcal{F}=\\{\textit{All linear functions of }Y\\}$. It readily
follows that,

$\displaystyle M_{L}(X,Y)=\textit{Maximum eigenvalue of }\Sigma$ (3.6)

where,

$\displaystyle\Sigma$
$\displaystyle=Var^{-\frac{1}{2}}(Y)Var(E(Y|X))Var^{-\frac{1}{2}}(Y)$
$\displaystyle=I-Var^{-\frac{1}{2}}(Y)E(Var(Y|X))Var^{-\frac{1}{2}}(Y)$

and $Var(W)$ is the variance-covariance matrix of $W\in R^{p}$.

This notion can be extended to $Y$ not fully dimensional by considering
singular values and $Y$ not in $L_{2}$ by, as for $C$, considering
$(G_{Y^{(1)}}(Y^{(1)}),...,G_{Y^{(p)}}(Y^{(p)}))$ where
$Y\equiv(Y^{(1)},...,Y^{(p)})$.

The definition becomes valid even if $G_{Y^{(j)}}(.)$ are not continuous.

The discrete case

If $(X,Y)$ has finite support
$x\in\\{x_{1},...x_{r}\\},y\in\\{y_{1},...y_{s}\\}$, then, $M_{L}$ defined by
(3.5) is the maximal canonical correlation between

$\displaystyle(1(X=x_{1}),.....,1(X=x_{r}))\textit{ and
}(1(Y=y_{1}),.....,1(Y=y_{s})).$

In particular, $M_{L}$ and $R$ coincide, since any function of
$(1(X=x_{1}),.....,1(X=x_{r}))$ can be written as a linear combination and
similarly for $Y$. Using (1.3) we get,

$\displaystyle M_{L}=\textit{maximal eigenvalue of
}{\|\sum_{j=1}^{s}{p_{aj}q_{jb}}\|_{r\times r}}$ (3.7)

(or the corresponding $s\times s$ matrix for $Y$).

Here, $p_{aj}\equiv P[Y=y_{j}|X=x_{a}],q_{jb}\equiv P[X=x_{b}|Y=y_{j}]$

There is an extensive literature on this measure and related topics in the
contingency table literature, see Goodman(1965), Gilula and Haberman Gilula
and Haberman (1995) for instance.

Shape constrained subspaces of $L_{2}(X)$ A natural extension to consider is
to situations where we require that the dependence parameter $M$ should be 1
if and only if $Y=h(X)$ for $h\in\mathcal{H}$, a convex cone in $L_{2}(X)$.
The simplest example, after $L_{2}$ itself which leads to $C$, is, the
isotonic case, taking $\mathcal{H}=\\{\textit{Monotone functions of }X\\}$. In
fact, a class of such parameters exist, namely $T_{S}$ and related ones such
as the normal scores correlation. All of these rely on the fact that, if
$F_{X}$ and $G_{Y}$ are continuous, $Y=h(X)$ for $h$ monotone iff
$F_{X}(X)=G_{Y}(Y)$ and the ordinary linear condition suffices. If we apply
(3.1) and define,

$\displaystyle C_{\mathcal{H}}(X,Y)\equiv
max\\{\rho^{2}(Y,h(X)):h\in\mathcal{H}\\},$

then, by [14] and (3.1),

$\displaystyle C_{\mathcal{H}}=\frac{Var(\Pi_{\mathcal{H}_{X}}(Y))}{Var(Y)}$
(3.8)

where,

$\displaystyle\Pi_{\mathcal{H}_{X}}(Y)\equiv\operatorname*{arg\,min}\\{E(Y-h(X))^{2}:h\in\mathcal{H}\\}$

exists uniquely, if $\mathcal{H}$ is a closed convex cone in $L_{2}(X)$.

Cao and Bickel show, in Cao and Bickel (2020), that, in the isotonic case,
unlike $C$, $C_{\mathcal{H}}$ can be estimated by plug in and the resulting
$\widehat{C}_{\mathcal{H}}$ is consistent and has interesting probabilistic
features. Unfortunately its behaviour as a test of independence is worse than
$\widehat{T}_{S}$ which has the same functional dependence properties that
$\widehat{C}_{\mathcal{H}}$ has.

For other $\mathcal{H}$, such as convex functions, plug in estimation seems
impossible, see [13], as it is for $C$, and as with Dette et al’s statistic is
likely to be consistent only under strong conditions on
$\Pi_{\mathcal{H}_{X}}(Y)$ as a function of $X$. We do not pursue these
applications further.

###  3.4 Estimation

The passage from (1.2) back to (1.1) is obscure. Here is a possible
clarification.

Chatterjee’s approach relies on the second formula in (1.2). Specifically, he
estimates $Var(1(Y\leq y)|X)$ by noting that, given $Y^{\prime}=y$, if there
were two independent variables $Y^{(1)},Y^{(2)}$ from the conditional
distribution of $Y$ given $X$, then,

$\displaystyle Var(1(Y\leq y)|X)=\frac{1}{2}E(1(Y^{(1)}\leq y)-1(Y^{(2)}\leq
y)^{2})$ $\displaystyle=\frac{1}{2}E(1(min(Y^{(1)},Y^{(2)})\leq y\leq
max(Y^{(1)},Y^{(2)}))$

As we have seen in Section 2.4, his final move is to take $Y_{(i)}$ and
$Y_{(i+1)}$ as proxies for $Y^{(1)},Y^{(2)}$, and use plug in, leading to
$\widehat{C}_{n}$.

Note that the same trick can also be applied to estimate
$Var\bigg{(}G_{Y}(Y)|F_{X}(X)\bigg{)}$, needed for $M(X,G_{Y}(Y))$ which
measures directly functional dependence of $Y$ on $X$.

We are led to,

$\displaystyle\widehat{M}\equiv
1-\frac{6}{n}\sum_{i=1}^{n-1}{(r_{i+1}-r_{i})^{2}}+O(\frac{1}{n})$
$\displaystyle=\frac{12}{n}\sum_{i=1}^{n-1}{\frac{r_{i}r_{i+1}}{n^{2}}}-3+O(\frac{1}{n})$
$\displaystyle=\widehat{T_{S}}^{(L)}+O(\frac{1}{n})$ (3.9)

The corresponding parameter,

$\displaystyle M(X,G_{Y}(Y))=\frac{Var(E(G_{Y}(Y)|X))}{Var(G_{Y}(Y))}$
$\displaystyle=1-\frac{E[Var(G_{Y}(Y)|X)]}{Var(G_{Y}(Y))}$

which is just $T_{S}^{(L)}$, the analogue of Spearman’s $\rho$.

It satisfies C) (Functional Dependence) but not B).

which can be rewritten as we did $C$ in (2.32) as

$\displaystyle=1-\frac{E_{F^{(2)}}\bigg{(}F_{2}(Y^{(1)})-F_{2}(Y^{(2)})\bigg{)}^{2}}{E\bigg{(}F_{2}(Y_{1})-F_{2}(Y_{2})\bigg{)}^{2}}$
(3.10)

This is equal if $F_{1}$ and $F_{2}$ are continuous to $T_{S}^{(L)}(F)$ which
is consistently estimated by $\widehat{T}_{S}^{(L)}$ given in (3.9).

Expression (3.10) suggests a class of statistics which have the same power and
functional dependence properties as $T_{S}^{(L)}(F)$ and $C$.

Given a monotone strictly increasing on $(0,1)$, define the statistic

$\displaystyle\widehat{T}_{a}^{(L)}=\frac{\frac{1}{n-1}\sum_{i=1}^{n-1}a(\frac{r_{i}}{n})a(\frac{r_{i+1}}{n})-\bar{a}^{2}}{\sigma^{2}(a)}$

where

$\displaystyle\bar{a}\equiv\frac{1}{n}\sum_{i=1}^{n}a(\frac{i}{n})$
$\displaystyle\sigma^{2}(a)\equiv\frac{1}{n}\sum_{i=1}^{n}\bigg{(}a(\frac{i}{n})-\bar{a}\bigg{)}^{2}$

$\widehat{T}_{a}^{(L)}$ under mild conditions given below consistently
estimates,

$\displaystyle
T_{a}^{(L)}(F)=1-\frac{E_{F^{(2)}}\bigg{(}a\big{(}F_{2}(Y^{(1)})\big{)}-a\big{(}F_{2}(Y^{(2)})\big{)}\bigg{)}^{2}}{E\bigg{(}a(F_{2}(Y_{1}))-a(F_{2}(Y_{2}))\bigg{)}^{2}}$

the analogues of objects each as the normal scores correlation (if
$a=\Phi^{-1}$)

We prove,

######  Theorem 3.1.

Assume $(U,V)\in[0,1]^{2}$ are such that $U\sim\mathcal{U}(0,1)$, $V$ given
$U=u$ has cdf $F(.|u)$ and $V$ has a continuous cdf,

$\displaystyle(i)\int|dF(v|u)-dF(v|u_{0})|\to 0\textit{ whenever }u\to u_{0}$
(i) can be interpreted as continuity of the map $u\to F(.|u)$ in
$\mathcal{P}_{0}$ variational norm on probabilities.
$\displaystyle(ii)q:[0,1]^{2}\to R,q\textit{ continuous}.$

Then,

$\displaystyle\frac{1}{n-1}\sum_{i=1}^{n-1}q\big{(}\frac{r_{i}}{n},\frac{r_{i+1}}{n}\big{)}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}q(v_{1},v_{2})dF(v_{1}|u)dF(v_{2}|u)du$
(3.11)

Proof of Theorem 3.1

Write,

$\displaystyle\frac{1}{n-1}\sum_{i=1}^{n-1}q\big{(}\frac{r_{i}}{n},\frac{r_{i+1}}{n}\big{)}=\frac{1}{n-1}\sum_{i=1}^{n-1}q\big{(}\widehat{F}_{2}(Y_{[i]}),\widehat{F}_{2}(Y_{[i+1]})\big{)}$

where $Y_{[i]}=Y_{j}$ iff $X_{j}=X_{(i)}$, where $X_{(i)}$ are the ordered
$X_{i}$ as usual. We can write $X_{(i)}\equiv F_{1}^{-1}(U_{(i)})$,
$Y_{(i)}\equiv F_{2}^{-1}(V_{[i]})$ with $(U_{i},V_{i})$ defined appropriately
by assumption $F_{1},F_{2}$ are continuous.

Note that given $X_{(1)},...,X_{(i)}$, the $Y_{[i]}$ are independent with
$Y_{[i]}\sim\mathcal{L}(Y|X=X_{(i)})$.

Next use Glivenko-Cantelli on the $Y_{1},...,Y_{n}$ and the uniform continuity
of $q$ to conclude that

$\displaystyle\frac{1}{n-1}\sum_{i=1}^{n-1}q(r_{i},r_{i+1})-\frac{1}{n-1}\sum_{i=1}^{n-1}q\big{(}V_{[i]},V_{[i+1]}\big{)}\stackrel{{\scriptstyle
a.s.}}{{\longrightarrow}}0$

But $V_{[1]},...,V_{[n]}$ is a double array of bounded conditionally
independent random variables.

Therefore, if $n=2M+2$, by the LLN for double arrays,

$\displaystyle\frac{1}{n-1}\sum_{k=0}^{n}\bigg{[}q\big{(}V_{[2k+1]},V_{[2k+2]}\big{)}-E\big{[}q\big{(}V_{[2k+1]},V_{[2k+2]}\big{)}|U_{(2k+1)},U_{(2k+2)}\big{]}\bigg{]}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}0$ (3.12)

Using $(V_{[2k]},V_{[2k+1]})$ in the same way we used
$(V_{[2k+1]},V_{[2k+2]})$ in (3.12) we obtain an analogous result.

Let

$\displaystyle Q(u_{1},u_{2})\equiv
E\bigg{[}q(V_{1},V_{2})|U_{1}=u_{1},U_{2}=u_{2}\bigg{]}$

If we add the first sum in (3.12) to the corresponding sum with
$(V_{[2k]},V_{[2k+1]})$, we see that the theorem is proved if we can show that

$\displaystyle\frac{1}{n-1}\sum_{i=1}^{n-1}Q(U_{(i)},U_{(i+1)})=\frac{1}{n}\sum_{i=1}^{n}Q(U_{(i)},U_{(i)})+o_{p}(1)$
(3.13)

Then we can just apply the LLN to

$\displaystyle\frac{1}{n}\sum_{i=1}^{n}Q(U_{(i)},U_{(i)})=\frac{1}{n}\sum_{i=1}^{n}Q(U_{i},U_{i})$

To prove (3.13) we write,

$\displaystyle Q(x_{1},x_{2})=\int
q(v_{1},v_{2})dF(v_{1}|x_{1})dF(v_{2}|x_{2})$

Then,

$\displaystyle Q(X_{(i)},X_{(i+1)})-Q(X_{(i)},X_{(i)})=\int\int
q(v_{1},v_{2})dF(v_{1}|X_{(i)})\big{(}dF(v_{2}|X_{(i+1)})-dF(v_{2}|X_{(i)})\big{)}$

We can now apply condition (i) using the boundedness of $q$ guaranteed by (ii)
to establish (3.13). ∎

Remarks:

1) Since $a(.)$ in $T_{a}^{(L)}$ is assumed continuous, consistency of
$T_{a}^{(L)}$ follows from Theorem 3.1, as does consistency of
$\widehat{C}_{n}$.

2) Condition (i) can be weakened using the methods of Chatterjee Chatterjee
(2020).

##  4 The best of both worlds

We will now argue that decoupling independence and functional dependence is
the ”right” thing to do. We want to construct measures of dependence $D(F)$
and corresponding empirical versions $\widehat{D}_{n}$, such that,
$\widehat{D}_{n}\to D(F)$ in probability for all cdf $F$ of $(X,Y)$ and these
quantities have the following properties:

  1. (i)

$0\leq D(F)\leq 1$

  2. (ii)

$D(F)=0$ iff $X$ and $Y$ are independent

  3. (iii)

$D(F)=1$ iff $Y=h(X)$ a.s. for some $h$

  4. (iv)

$\widehat{D}_{n}$ based tests have ”optimal” local power properties against
$H$: $X$ and $Y$ are independent.

Let $T_{I}(F)$ be any measure of dependence such that $0\leq T_{I}(F)\leq 1$,
$T_{I}(F)=0$ iff $X$ and $Y$ are independent. Let $\widehat{T_{I}}$ be an
associated statistic which is consistent whatever be $F$, and has, as a test
statistic, any desired local power properties. From our discussion this might
be power against any local alternative for which $\dot{l}_{0}(X,Y)\neq
a(X)+b(Y)$, $a,b\neq 0$. Or it might be particularly good local power
behaviour in some directions as in Bickel, Ritov and, Stoker (Bickel et al.
(2006)). Suppose also that $T_{I}(F)<1$, for all $F$. This parameter may now
naturally be coupled to $T_{FD}(F)$ such that $T_{FD}(F)=1$ iff $Y=h(X)$ a.s.,
$\widehat{T_{FD}}$, is consistent for $T_{FD}(F)$, and $H\Rightarrow
T_{FD}(F)=0$.

Let,

$\displaystyle D(F)$ $\displaystyle=T_{I}(F)\textit{ if }0\leq
T_{FD}(F)\leq\frac{1}{2}$ $\displaystyle=T_{FD}(F)\textit{ otherwise.}$

Let $\widehat{D}$ be similarly defined in terms of $\widehat{T_{I}}$ and
$T_{FD}(\widehat{F})$.

Note that $\frac{1}{2}$ may be replaced by any number $<1$.

Then, clearly,

$\displaystyle(i)P_{0}[\widehat{D}=\widehat{T_{I}}]\to 1$
$\displaystyle(ii)\widehat{D}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}0\Rightarrow H$
$\displaystyle(iii)\widehat{D}\stackrel{{\scriptstyle p}}{{\longrightarrow}}1$
$\displaystyle\textit{iff }Y=h(X)\textit{ a.s.}(FD)$

(i)-(iii) are argued as follows.

(i) $H\Rightarrow\widehat{T}_{I}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}0$ and $\widehat{T}_{FD}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}0\Rightarrow\widehat{D}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}0$.

(ii)$\widehat{D}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}0\Rightarrow\widehat{T}_{I}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}0$ AND $\widehat{T}_{FD}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}0$.

The first implication implies $H$.

The second implies $P[\widehat{D}=\widehat{T}_{I}]\to 1$ by definition, so
that the first case holds as well.

(iii) a)$FD\Rightarrow\widehat{T}_{FD}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}1\Rightarrow P[\widehat{D}=\widehat{T}_{FD}]\to
1\Rightarrow\widehat{D}\stackrel{{\scriptstyle p}}{{\longrightarrow}}1$,

b) $\widehat{D}\stackrel{{\scriptstyle p}}{{\longrightarrow}}1\Rightarrow
P[\widehat{D}=\widehat{T}_{FD}]\to 1\Rightarrow FD$

Both a) and b) hold by consistency and the assumption that $T_{I}(F)<1$.

Thus $D,\widehat{D}$ have the properties we want. For instance, we can take
$\widehat{T}_{I}=\widehat{T}_{BKR}$.

and $\widehat{T}_{FD}=\widehat{C}_{n}$ or $\widehat{T}_{S}^{(L)}$. Or we could
add desired local power properties by taking

$\displaystyle\widehat{T}_{I}=max\\{\widehat{T}_{BKR},\widehat{T}_{BKR}^{(L)}\\}.$

We use Lemma A2, establishing $T_{BKR}^{(F)}<1$. The same property applies to
$T^{(L)}_{BKR}$ by the same argument. Clearly,
$H\Rightarrow\widehat{T}_{S}^{(L)}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}0$ and FD holds iff
$\widehat{T}_{S}^{(L)}\stackrel{{\scriptstyle p}}{{\longrightarrow}}1$.

$\mathbb{Discussion:}$ As Lin and Han (2021) show, it is possible by going to
parameters which depend on $F^{(m)},m\to\infty$ to obtain extensions of
$\widehat{C_{n}}$ which have good minimax properties beyond the
$n^{-\frac{1}{4}}$ detection barrier, as they define it.

This could be done more generally with parameters like $D(F)$ but we do not
pursue this or generalizations to other spaces than $R$, as done by Deb,
Ghosal and Sen (2020).

##  Appendix A Lemma A.1

######  Lemma A.1.

If $\dot{l}_{0}\neq 0$ satisfies the hypothesis of Theorem 2.1, then $C(.)$
given in (2.27) is not 0.

$\mathbb{Proof:}$

By definition,

$\displaystyle T_{n}(.)\stackrel{{\scriptstyle
d}}{{\longrightarrow}}T(.)+tC(.)$

where,

$\displaystyle
C(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}})=\int\Psi(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}},\hbox{\oalign{$\bm{w}$\crcr$\scriptscriptstyle\bm{\sim}$}})\dot{l}_{0}(\hbox{\oalign{$\bm{w}$\crcr$\scriptscriptstyle\bm{\sim}$}})d\hbox{\oalign{$\bm{w}$\crcr$\scriptscriptstyle\bm{\sim}$}}$
(A.11)

and

$\displaystyle\Psi(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}},\hbox{\oalign{$\bm{w}$\crcr$\scriptscriptstyle\bm{\sim}$}})=1(\hbox{\oalign{$\bm{w}$\crcr$\scriptscriptstyle\bm{\sim}$}}\leq\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}})-w_{1}1(w_{2}\leq
u_{2})-w_{2}1(w_{1}\leq u_{1})+w_{1}w_{2}$ (A.20)

Here,
$\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}}=(u_{1},u_{2})$,
$\hbox{\oalign{$\bm{w}$\crcr$\scriptscriptstyle\bm{\sim}$}}=(w_{1},w_{2})$,
$\hbox{\oalign{$\bm{w}$\crcr$\scriptscriptstyle\bm{\sim}$}}\leq\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}}$
iff $w_{j}\leq u_{j},j=1,2$.

Rewriting,

$\displaystyle
C(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}})=E\Psi(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}},U,V)\dot{l}_{0}(U,V)$
(A.25)

for $U,V$ iid $\mathcal{U}(0,1)$. By construction, for all  $\bm{u}$
$\scriptscriptstyle\bm{\sim}$ ,

$\displaystyle
E(\Psi(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}},U,V)|U)=E(\Psi(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}},U,V)|V)=0$
(A.30)

Therefore, if
$C(\hbox{\oalign{$\bm{u}$\crcr$\scriptscriptstyle\bm{\sim}$}})=0$ for all
$\bm{u}$ $\scriptscriptstyle\bm{\sim}$ , $Ef(U,V)\dot{l}_{0}(U,V)=0$ for all
$f\in\Lambda_{1}^{\perp}$.

But, $\dot{l}_{0}\in\Lambda_{1}^{\perp}$.

The lemma follows. ∎

######  Lemma A.2.

$T_{BKR}(F)<1$ for all $F$.

$\mathbb{Proof:}$ Suppose $F_{1},F_{2}$ are continuous. Then $T_{BKR}(F)$ is
given by (2.2).

By (2.2) we an take $F_{j}(u)=u$ for $j=1,2$.

Since $|F(u,v)-uv|\leq 1$ for all $u,v$, we must have,

$\displaystyle|F_{m}(u,v)-uv|\to 1$

in measure on $[0,1]^{2}$. Since the sequence $\\{F_{m}\\}$ is tight, there
exists $\\{m_{k}\\}$, $F_{0}$ such that $F_{m_{k}}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}F_{0}$ and

$\displaystyle|F_{0}(u,v)-uv|=1\textit{ a.e. }(u,v)$

equivalent to,

$\displaystyle F_{0}(u,v)=1+uv\textit{ or }uv-1.$

This is impossible in the interior of $[0,1]^{2}$. The result follows for
$F_{1},F_{2}$ continuous.

In general, write,

$\displaystyle
T_{BKR}(F)=\int_{0}^{1}\int_{0}^{1}\bigg{(}F\big{(}F_{1}^{-1}(u),F_{2}^{-1}(v)\big{)}-F_{1}\big{(}F_{1}^{-1}(u)\big{)}F_{2}\big{(}F_{2}^{-1}(v)\big{)}\bigg{)}^{2}dudv$

The argument given before holds if $T_{BKR}=1$,

$\displaystyle
F_{1}\big{(}F_{1}^{-1}(u)\big{)}F_{2}\big{(}F_{2}^{-1}(v)\big{)}=0\textit{ and
}F\big{(}F_{1}^{-1}(u),F_{2}^{-1}(v)\big{)}=1$ (A.31)

or conversely for almost all $u,v$.

But if $(X,Y)$ takes on only 1 value,

$\displaystyle F_{1}F_{2}=F\textit{ a.e.}$

While if $(X,Y)$ takes on more than 1 value but is not continuous, (A.5) is
impossible since $F(.,.)$ takes on more than 1 value. In either case,m the
lemma holds. ∎

##  Appendix B Proof of Lemma 2.2

We begin with an identity based on Fubini’s theorem. For any finite signed
measure $\mu$ on $[0,1]^{2}$ if $\widetilde{A}(s)=A(1)-A(s)$, $\widetilde{B}$
is similarly defined, and $\mu(s,t)=\int_{0}^{s}\int_{0}^{t}d\mu(u,v)$, then

$\displaystyle\int_{0}^{1}\int_{0}^{1}\bar{\widetilde{A}}(s)\bar{\widetilde{B}}(t)d\mu(s,t)=\int_{0}^{1}\int_{0}^{1}a(s)b(t)\mu(s,t)dsdt$
(B.1)

Note that, $\bar{\widetilde{A}}(s)=-\bar{A}(s)$.

Then, if $C(u,v)=uv$,

$\displaystyle\frac{1}{\sqrt{n}}\sum_{k=1}^{n}\bar{A}(\frac{R_{k}}{n})\bar{B}(\frac{S_{k}}{n})=\sqrt{n}\int_{0}^{1}\int_{0}^{1}\bar{\widetilde{A}}(u)\bar{\widetilde{B}}(v)d(\widehat{C}-C)(u,v)$
(B.2)

since $\int_{0}^{1}\bar{A}(u)du=\int_{0}^{1}\bar{B}(v)dv=0$.

Apply (A.5) with $\mu=\widehat{C}-C$ to get,

$\displaystyle\int_{0}^{1}\int_{0}^{1}\bar{\widetilde{A}}(u)\bar{\widetilde{B}}(v)d(\widehat{C}-C)(u,v)=\int_{0}^{1}\int_{0}^{1}a(s)b(t)\sqrt{n}(\widehat{C}-C)(s,t)dsdt$

Apply Proposition 2.1 to get,

$\displaystyle\int_{0}^{1}\int_{0}^{1}a(s)b(t)\sqrt{n}(\widehat{C}-C)(s,t)dsdt=\int_{0}^{1}\int_{0}^{1}a(s)b(t)(E_{n}(s,t)-sE_{n2}(t)-tE_{n1}(s))dsdt+o_{p}(1)$
(B.3)

since $sup\\{|E_{n}(s,t)|\\}=O_{p}(1)$.

Apply (B.1) again to $E_{n}(s,t)-sE_{n2}(t)-tE_{n1}(s)$, which is of bounded
variation for each n to finally obtain,

$\displaystyle\int_{0}^{1}\int_{0}^{1}a(s)b(t)(E_{n}(s,t)-sE_{n2}(t)-tE_{n1}(s))dsdt=$
$\displaystyle=\int_{0}^{1}\int_{0}^{1}\bar{\widetilde{A}}(s)\bar{\widetilde{B}}(t)dE_{n}(s,t)$

since $\int_{0}^{1}ta(t)dt=\int_{0}^{1}\bar{A}(t)dt$. The lemma follows. ∎

##  Appendix C Proof of Lemma 2.3

Let $B_{n}=\sigma$ field generated by $U_{1},...,U_{n}$.

We can then write $A_{n}=A_{n1}+A_{n2}$, where

$\displaystyle A_{n1}\equiv E(A_{n}|B_{n})$
$\displaystyle=\frac{1}{\sqrt{n}}\sum_{k=1}^{n}\int_{0}^{1}a(U_{k},v)dv$
$\displaystyle=\frac{1}{\sqrt{n}}\sum_{k=1}^{n}a_{1}(U_{k})$

$\displaystyle A_{n2}\equiv A_{n}-A_{n1}$

Given $B_{n}$, $A_{n2}$ is a sum of n independent, non identically distributed
variables with mean 0 and

$\displaystyle
Var(A_{n2}|B_{n})=\frac{1}{n}\sum_{k=1}^{n}Var(a(U_{k},V_{k})|V_{k})$

By the Lindeberg Feller theorem and the LOL,

$\displaystyle Var(A_{n2}|B_{n})\stackrel{{\scriptstyle
a.s.}}{{\longrightarrow}}E\\{Var[a(U_{1},V_{1})|U_{1}]\\}\equiv\sigma_{2}^{2}$

and

$\displaystyle A_{n2}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}Z_{0}^{(2)}\sim N(0,\sigma_{2}^{2})$

$A_{n1}$ is uncorrelated with $A_{n2}$ and by the CLT,

$\displaystyle A_{n1}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}Z_{0}^{(1)}\sim N(0,\sigma_{1}^{2}),\sigma_{1}^{2}\equiv
Ea_{1}^{2}(U_{1})$

Therefore,

$\displaystyle(A_{n1},A_{n2})\stackrel{{\scriptstyle
d}}{{\longrightarrow}}(Z_{0}^{(1)},Z_{0}^{(2)})$

where $Z_{0}^{(1)},Z_{0}^{(2)}$ are independent $N(0,\sigma_{1}^{2})$,
$N(0,\sigma_{2}^{2})$.

Consider $h^{(1)}(v_{1},v_{2})$, and write

$\displaystyle
H_{n}^{(1)}=\frac{1}{\sqrt{n}}\sum_{k=1}^{n-1}h^{(1)}(W_{k},W_{k+1})$

where $W_{k}=V_{D_{k}}$ as before are iid $\mathcal{U}(0,1)$ given $B_{n}$. By
Orey’s (1958) Orey (1958) theorem for 2 dependent random vairable,
conditionally,

$\displaystyle H_{n}^{(1)}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}Z_{1}\sim N(0,Var(Z_{1}))$

with $Var(Z_{1})$ as specified in (2.34).

The same applies to $H_{n}^{(2)}$ and the joint distribution is as specified.
We can put $(A_{n2},H_{n}^{(1)},H_{n}^{(2)})$ together to obtain joint
convergence given $B_{n}$. Since the limiting covariance matrix is independent
of $B_{n}$, we can add $A_{n1}$ as asymptotically independent of
$(A_{n2},H_{n}^{(1)},H_{n}^{(2)})$. Since the covariances of the limits are in
all cases the limit of the unconditional covariances the lemma follows on
direct computation of the latter.

∎

##  Appendix D Proof of Proposition 2.4

Consider for $(X_{i},Y_{i})$ iid $h_{0}$, and apply Taylor’s theorem to

$\displaystyle\sum_{i=1}^{n}\big{[}l_{\theta}(X_{i},Y_{i})-l_{0}(X_{i},Y_{i})\big{]}$
$\displaystyle=\theta\sum_{i=1}^{n}\big{(}a(X_{i})+b(Y_{i})\big{)}+\frac{\theta^{2}}{2}\sum_{i=1}^{n}\frac{\partial^{2}l_{\theta}}{\partial\theta^{2}}\bigg{|}_{0}(X_{i},Y_{i})+\frac{\theta^{3}}{6}R_{n}(\theta)$
(D.1)

By assumption, for $|\theta|\leq\frac{M}{\sqrt{n}}$,

$\displaystyle|R_{n}(\theta)|\leq\frac{M^{3}}{6n^{\frac{1}{2}}}K=o(1)$ (D.2)

On the other hand, since $Ea(X)=Eb(Y)=0$, $a,b$ bounded, under $H$,

$\displaystyle\theta_{n}\sum_{i=1}^{n}\big{(}a(X_{i})+b(Y_{i})\big{)}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}Z$ (D.3)

if $\sqrt{n}\theta_{n}\to t,\sigma^{2}\equiv
E_{0}\big{(}a(X)+b(Y)\big{)}^{2},Z\sim N(0,t^{2}\sigma^{2})$.

By a standard argument, under our conditions,

$\displaystyle
E_{0}\big{(}\frac{\partial^{2}}{\partial\theta^{2}}l_{\theta}(X_{i},Y_{i})\big{|}_{0}\big{)}=-\sigma^{2}$

Thereforem, by the LLN, if $\sqrt{n}\theta_{n}\to t$,

$\displaystyle\frac{\theta_{n}^{2}}{2}\sum_{i=1}^{n}\frac{\partial^{2}}{\partial\theta^{2}}l_{\theta}(X_{i},Y_{i})\big{|}_{0}\stackrel{{\scriptstyle
a.s}}{{\longrightarrow}}-\frac{t^{2}}{2}\sigma^{2}$ (D.4)

The same argument applies using only the boundedness of $a,b$ to
$\widetilde{h}(X_{i},Y_{i})$. Write,

$\displaystyle\Delta_{n}^{(1)}(\theta)=E_{0}\big{|}\prod_{i=1}^{n}\frac{h_{\theta}(X_{i},Y_{i})}{h_{0}(X_{i},Y_{i})}-exp\\{\theta\sum_{i=1}^{n}(a(X_{i})+b(Y_{i}))-\frac{\theta^{2}}{2}\sigma^{2}\\}\big{|}$
(D.5)

and define $\Delta_{n}^{(2)}(\theta)$ similarly for the
$\widetilde{h}(X_{i},Y_{i})$.

Then,

$\displaystyle\Delta_{n}(\theta)\leq\Delta_{n}^{(1)}(\theta)+\Delta_{n}^{(2)}(\theta)$

We can write, letting $A_{n}(.)$ and $B_{n}(.)$ be the logs of the expressions
in (D.5),

$\displaystyle\Delta_{n}^{(1)}(\theta)\equiv
E_{0}|e^{A_{n}(\theta)}-e^{B_{n}(\theta)}|$

By (D.2),

$\displaystyle|A_{n}(\theta)-B_{n}(\theta)|=o_{p}(1)$ (D.6)

Moreover, by (D.3) and (D.4),

$\displaystyle e^{B_{n}(\theta)}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}e^{A_{t}}$

where $A_{t}=Z-\frac{t^{2}\sigma^{2}}{2}$ and $Ee^{A_{t}}=1$.

Since $E_{0}e^{A_{n}(\theta)}=1$ for all $n$, we can conclude by uniform
integrability of the $A_{n}(\theta)$ that,

$\displaystyle E_{0}e^{A_{n}(\theta)}|1-e^{B_{n}(\theta)-A_{n}(\theta)}|\to 0$

The same argument applies to $\Delta_{n}^{(2)}(\theta)$ and we can conclude
that if $\sqrt{n}\theta_{n}\to t$,

$\displaystyle\Delta_{n}(\theta_{n})\to 0$

Finally, we note that if
$sup\\{\Delta_{n}(\theta):|\theta|\leq\frac{M}{\sqrt{n}}\\}$ does not tend to
0, there must exist $\theta_{n}$ with $|\sqrt{n}\theta_{n}|\to|t|\leq M$ for
which $\Delta_{n}(\theta_{n})$ does not tend to 0, a contradiction.

∎

## Acknowledgements

This work is based in part on joint work with Sky Cao Cao and Bickel (2020).
His contribution to the current paper amounts to coauthorship. However, any
errors are the responsibility of the author.

We thank Sourav Chatterjee for facilitating this collaboration, as well as for
helpful conversations. We thank Holger Dette for pointing out an important
reference. We thank Hongjian Shi and Mathias Drton for comments on the local
power calculations. In particular, we thank Fang Han also for valuable
comments on various parts of the paper.

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