#  Nash implementation in a many-to-one matching market ††thanks: We thank
Agustín Bonifacio and Pablo Neme for their very detailed comments. We
acknowledge financial support from UNSL through grants 032016 and 030320, from
Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET) through
the grant PIP 112-200801-00655, and from Agencia Nacional de Promoción
Científica y Tecnológica through grant PICT 2017-2355.

Noelia Juarez Instituto de Matemática Aplicada San Luis (UNSL-CONICET) and
Departamento de Matemática, Universidad Nacional de San Luis, San Luis,
Argentina. Emails<EMAIL_ADDRESS>(N. Juarez<EMAIL_ADDRESS>(P. B. Manasero) and<EMAIL_ADDRESS>(J. Oviedo). Paola B.
Manasero22footnotemark: 2 Jorge Oviedo22footnotemark: 2

###### Abstract

In a many-to-one matching market, we analyze the matching game induced by a
stable rule when firms’ choice function satisfy substitutability. We show that
any stable rule implements the individually rational correspondence in Nash
equilibrium when both sides of the market play strategically. Moreover, when
only workers play strategically and firms’ choice functions satisfy the law of
aggregate demand, we show that the firm-optimal stable rule implements the
stable correspondence in Nash equilibrium.

_JEL classification:_ C78, D47.

_Keywords:_ Stable matchings, Nash equilibrium, substitutable preferences,
matching game.

##  1 Introduction

In this paper, we study a many-to-one matching market in which agents on one
side of the market (which we call _firms_) must be assigned to subsets of
agents on the other side of the market (which we call _workers_), and the only
requirement on subsets of workers that each firm’s choice function must
satisfy is substitutability. Kelso and Crawford (1982) introduced this
condition, which is the weakest requirement on firms’ choice functions in
order to guarantee the existence of stable matchings. A firm has substitutable
choice functions if it wants to continue hiring a worker even if other workers
become unavailable.

In centralized markets, a board needs to collect the preferences and choice
functions of all agents in order to produce a stable matching. Normally,
agents are expected to behave strategically by not revealing their true
preferences or their true choice functions in order to benefit themselves.
When this is the case, the matching market becomes a matching game.

A stable rule is a function that associates each stated strategy profile to a
stable matching under this stated profile. To evaluate such matchings workers,
and firms use their true preferences and their true choice functions,
respectively. A market and a stable rule induce a matching game. In this game,
the set of strategies for each worker is the set of all possible preferences
that she could state. Similarly, the set of strategies for each firm is the
set of all possible choice functions that it could state.

In this paper, the equilibrium concept we focus on is Nash equilibrium. In a
Nash equilibrium, no agent improves by deviating from its initial chosen
strategy, assuming the other agents keep their strategies unchanged.

It is well known, for matching markets, that there is no stable rule for which
truth-telling is a dominant strategy for all agents (see Dubins and Freedman,
1981; Roth, 1982, 1985; Sotomayor, 1996, 2012; Martínez et al., 2004; Manasero
and Oviedo, 2022, among others). That is, given the true preferences and a
stable rule, at least one agent might benefit from misrepresenting her
preferences regardless of what the rest of the agents state. Thus, stable
matchings cannot be reached through dominant “truth-telling equilibria". The
stability of equilibrium solutions under true preferences is expected to be
affected when agents behave strategically.

Given a market, the question that arises is what are the rules that induce a
matching game that allows us to implement stable matchings in Nash
equilibrium. First, we study a matching game in which the players are all the
agents. We show that any stable rule implements, in Nash equilibrium, the
individually rational matchings. Second, to implement stable matchings, we
focus on another matching game. In some markets, such as school choice or
labor markets, institutions are legally required to declare their true
preferences (priorities or choice functions), i.e., their preferences are
public. In these cases, individuals (students or workers) are expected to
manipulate their preference lists to their advantage. This situation leads us
to consider a matching game in which the players are only the workers.

Furthermore, when firms’ choice functions satisfy, in addition to
substitutability, the “law of aggregate demand" (LAD, from now on),111This
property is first studied by Alkan (2002) under the name of “cardinal
monotonicity”. See also Hatfield and Milgrom (2005). we show that the firm-
optimal stable rule implements, in Nash equilibrium, the stable matchings.

The contribution of this paper is to generalize the approach presented by
Sotomayor (2008, 2012) for the many-to-one matching market with responsive
preferences (a more restrictive requirement than substitutability) to
substitutable choice functions.

There is an extensive literature that focuses on studying the implementation
of rules using Maskin’s results as the main tool (see Maskin, 1977, 1999; Kara
and Sönmez, 1996, 1997; Ehlers, 2004; Haake and Klaus, 2009, among others). In
addition, to study implementation, the aforementioned authors analyze the
relationship between stability, monotonicity, individual rationality, and
Pareto efficiency. Maskin (1977, 1999) shows that a monotonicity condition
(Maskin’s monotonicity) is necessary for a rule to be Nash implementable.
Also, he shows that Maskin’s monotonicity and no veto power together are
sufficient conditions for implementability. It is important to highlight that,
unlike previous work, our results cannot be obtained through Maskin’s
implementation result since, although stable rules satisfy Maskin’s
monotonicity, they do not satisfy no veto power (see Kara and Sönmez, 1996,
for more details) .

For the one-to-one and many-to-one markets with responsive preferences, Kara
and Sönmez (1996, 1997) show that the stable rules are Nash implementable.
Ehlers (2004) obtains positive implementation results in one-to-one markets
when agents are allowed to have weak preferences. In a many-to-one market with
contracts, Haake and Klaus (2009) show that stable rules are Maskin monotonic
and implementable. All of the above articles demonstrate the implementability
of stable rules using some implementation conditions; e.g. monotonicity Maskin
(1999), essential monotonicity Yamato (1992) or the implementability condition
of Moore and Repullo (1990). In contrast, we focus on studying a game and
identifying the strategies that are Nash equilibria of the game and that allow
us to implement stable solutions.

In a one-to-one market, Alcalde (1996) studies the design of specific
mechanisms to implement stable solutions. He introduces two types of
mechanisms. One implements the set of all stable matchings in undominated Nash
equilibria; the other implements the optimal stable matching for one of the
two sides of the market via dominance resolvability. This last mechanism is
the result of the classic algorithm in matching theory, the Gale-Shapley
mechanism.

Unlike us, Alcalde and Romero Medina (2000) analyze a two-stage game in a
many-to-one market under substitutability. They focus on studying the notions
of subgame perfect Nash equilibrium and strong subgame perfect Nash
equilibrium. Under these notions of equilibrium, they implement the stable
correspondence. Thus, their approach is tangential to ours.

The rest of the paper is organized as follows. Section 2 presents the model
and some preliminaries. All the results of the paper are presented in Section
3. First, we show that any stable rule implements the individually rational
correspondence in Nash equilibrium. Second, assuming that only workers play
strategically, we show that the firm-optimal stable rule implements, in Nash
equilibrium, the stable correspondence when firms’ preferences satisfy
substitutability and LAD. Finally, concluding remarks are gathered in Section
4.

##  2 Model and preliminaries

We consider a many-to-one matching market where there are two disjoint sets of
agents: the set of workers $W$ and the set of firms $F$. Each worker $w\in W$
has a strict preference relation $P_{w}$ over the individual firms and the
prospect of being unmatched, denoted by $\emptyset$. For each $w\in W$,
$R_{w}$ is the weak preference over $F$ associated with $P_{w}$. Each firm
$f\in F$ has a choice function $C_{f}$ over the set of all subsets of $W$ that
satisfies substitutability, i.e., for $S^{\prime}\subseteq S\subseteq W$, we
have $C_{f}(S)\cap S^{\prime}\subseteq C_{f}(S^{\prime})$.222Substitutability
is equivalent to the following: for each $w\in W$ and each $S\subseteq W$ such
that $w\in S$, $w\in C_{f}\left(S\right)$ implies that $w\in
C_{f}\left(S^{\prime}\cup\\{w\\}\right)$ for each $S^{\prime}\subseteq S.$ In
addition, we assume that $C_{f}$ satisfies $C_{f}(S^{\prime})=C_{f}(S)$
whenever $C_{f}(S)\subseteq S^{\prime}\subseteq S\subseteq W.$ This property
is known in the literature as consistency. If $C_{f}$ satisfies
substitutability and consistency, then it also satisfies

$C_{f}\left(S\cup S^{\prime}\right)=C_{f}\left(C_{f}\left(S\right)\cup
S^{\prime}\right)$ (1)

for each pair of subsets $S$ and $S^{\prime}$ of $W$.333This property is known
in the literature as _path independence_ (see Alkan, 2002). Let
$P=(P_{w})_{w\in W}$ be the preference profile for all workers, and let
$C=(C_{f})_{f\in F}$ be the profile of choice functions for all firms. A
_many-to-one matching market_ is denoted by $(W,F,P,C).$ Since the sets $F$
and $W$ are kept fixed throughout the paper, we often identify the market
$(W,F,P,C)$ with the profile $(P,C)$. Given the profile $P$, we consider that
each $w\in W$ may misrepresent her preferences $P_{w}$, by some preference
$P^{\prime}_{w}$. We denote by $\left(P^{\prime}_{w},P_{-w}\right)$ the new
profile with the misreport, where $P_{-w}$ is the subprofile obtained by
removing $P_{w}$ from $P$ . We denote by $\mathcal{P}$ the set of all worker
preference profiles. We consider that each $f\in F$ may misrepresent her
choice function $C_{f}$, by some choice function $C^{\prime}_{f}$. We denote
by $\mathcal{C}$ the set of all firms’ choice functions profiles. Throughout
this paper, we assume that $C$ is substitutable for each $C\in\mathcal{C}$.

###### Definition 1

A matching $\mu$ is a function from $W\cup F$ into $2^{W\cup F}$ such that,
for each $w\in W$ and each $f\in F$:

  1. (i)

$\mu(w)\subseteq F$ with $|\mu(w)|\leq 1.$

  2. (ii)

$\mu(f)\subseteq W$.

  3. (iii)

$w\in\mu(f)$ if and only if $\mu(w)=\\{f\\}$.444Usually, we will omit the
curly brackets. For instance, instead of condition (iii) we will write:
“$w\in\mu(f)$ if and only if $\mu(w)=f$”. No confusion will arise.

Let $\mathcal{M}$ be the set of all matchings. An agent $a\in W\cup F$ is
matched if $\mu(a)\neq\emptyset$, otherwise $a$ is unmatched. A matching $\mu$
is blocked by a worker $\boldsymbol{w}$ if $\emptyset P_{w}\mu(w)$; that is,
worker $w$ would rather be unemployed than work for firm $\mu(w)$. Similarly,
$\mu$ is blocked by a firm $\boldsymbol{f}$ if $\mu(f)\neq
C_{f}\left(\mu(f)\right)$; that is, firm $f$ wants to fire some workers in
$\mu(f)$. A matching is individually rational if it is not blocked by any
individual agent. The set of individually rational matchings for market
$(P,C)$ is denoted by $\boldsymbol{I(P,C)}.$

A matching $\mu$ is blocked by a firm-worker pair $\boldsymbol{(f,w)}$ if
$w\in C_{f}\left(\mu(f)\cup\\{w\\}\right),$ and $fP_{w}\mu(w)$; that is, if
they are not matched by $\mu$, firm $f$ wants to hire $w$, and worker $w$
prefers firm $f$ to $\mu(w)$. A matching $\mu$ is stable if it is individually
rational and it is not blocked by any firm-worker pair. The set of stable
matchings for market $(P,C)$ is denoted by $\boldsymbol{S(P,C)}.$

For each $f\in F$, the choice function $C_{f}$ induces a preference binary
relation as follows: given two sets of workers $S,T\subseteq W$, we write
$\boldsymbol{S\succeq_{f}T}$ if and only if $S=C_{f}\left(S\cup
T\right)$.555This relation $\succeq$ was used in Blair (1988). Furthermore,
given two matchings $\mu$ and $\mu^{\prime}$, we write
$\boldsymbol{\mu\succeq_{F}\mu^{\prime}},$ whenever
$\mu(f)\succeq_{f}\mu^{\prime}(f)$ for each $f\in F.$ Similarly, we write
$\boldsymbol{\mu R_{W}\mu^{\prime}},$ whenever $\mu(w)R_{w}\mu^{\prime}(w)$
for each $w\in W.$

The set of stable matchings under substitutable choice functions is very well
structured. Blair (1988) proves that this set has two lattice structures, one
concerning $\succeq_{F}$ and the other one concerning $R_{W}$. Furthermore, it
contains two distinctive matchings: the firm-optimal stable matching under
$(P,C),$ $\mu_{F}(P,C)$, and the worker-optimal stable matching under $(P,C)$,
$\mu_{W}(P,C)$. The matching $\mu_{F}$ is unanimously considered by all firms
to be the best among all stable matchings and $\mu_{W}$ is unanimously
considered by all workers to be the best among all stable matchings (see Roth,
1984; Blair, 1988, for more details).

In a seminal paper, Gale and Shapley (1962) introduce the Deferred Acceptance
(_DA_ , from now on) algorithm, which computes the optimal stable matching for
one side of the market. Later, the _DA_ algorithm is adapted to the many-to-
one market when firms preferences are substitutable, by Roth and Sotomayor
(1990).

##  3 Nash Implementation

###  3.1 The individually rational correspondence

Before introducing the matching game, we need some standard terms and
notation.

A social choice correspondence $\Psi:\mathcal{P}\times\mathcal{C}\rightarrow
2^{\mathcal{M}}$ selects, for each market
$(P^{\prime},C^{\prime})\in\mathcal{P}\times\mathcal{C}$, a subset of
matchings $\Psi(P^{\prime},C^{\prime})\subseteq\mathcal{M}.$ The individually
rational correspondence $\mathcal{I}:\mathcal{P}\times\mathcal{C}\rightarrow
2^{\mathcal{M}}$ selects for each market $(P^{\prime},C^{\prime})$, the set of
individually rational matchings $I(P^{\prime},C^{\prime})$. In this context,
$P^{\prime}\in\mathcal{P}$ is a strategy profile for all workers where for
each $w\in W$, $P^{\prime}_{w}$ is $w$’s strategy. Moreover
$C^{\prime}\in\mathcal{C}$ is a strategy profile for all firms, where for each
$f\in F$, $C^{\prime}_{f}$ is the $f$’s strategy. A strategy profile for all
agents is $(P^{\prime},C^{\prime})\in\mathcal{P}\times\mathcal{C}.$ A
(matching) rule is a function
$\psi:\mathcal{P}\times\mathcal{C}\rightarrow\mathcal{M}$ that selects for
each strategy profile $(P^{\prime},C^{\prime})\in\mathcal{P}\times\mathcal{C}$
a matching $\psi(P^{\prime},C^{\prime})\in\mathcal{M}.$ A matching rule $\psi$
is stable if $\psi(P^{\prime},C^{\prime})\in S(P^{\prime},C^{\prime})$ for
each $(P^{\prime},C^{\prime})\in\mathcal{P}\times\mathcal{C}.$

Given $(P,C)\in\mathcal{P}\times\mathcal{C}$ and a stable rule $\psi$ the
(matching) game induced by $(P,C)$ and $\psi$ is denoted by $(\psi,P,C).$ A
strategy profile $(P^{\prime},C^{\prime})$ is a Nash equilibrium of
$(\psi,P,C)$, if no agent can achieve a better outcome deviating from her
strategy, assuming that the other agents do not deviate from the strategy
profile $(P^{\prime},C^{\prime})$. Formally,

###### Definition 2

Let be $(\psi,P,C)$ the game induced by $(P,C)$ and $\psi$. A strategy profile
$(P^{\prime},C^{\prime})$ is a Nash equilibrium of $(\psi,P,C)$ if for each
$w\in W$,
$\psi(P^{\prime},C^{\prime})(w)R_{w}\psi(\widehat{P}_{w},P^{\prime}_{-w},C^{\prime})(w)$
for each strategy $\widehat{P}_{w}$ of $w$, and for each $f\in F$,
$\psi(P^{\prime},C^{\prime})(f)\succeq_{f}\psi(P^{\prime},\widehat{C}_{f},C^{\prime}_{-f})(f)$
for each strategy $\widehat{C}_{f}$ of $f$.

###### Definition 3

We say that the game $(\psi,P,C)$ implements the social choice correspondence
$\Psi$ in Nash equilibrium if,

  1. (i)

for each Nash equilibrium $(P^{\prime},C^{\prime})$ of $(\psi,P,C)$,
$\psi(P^{\prime},C^{\prime})\in\Psi(P,C)$,

  2. (ii)

for each matching $\mu\in\Psi(P,C)$ there is a Nash equilibrium
$(P^{\prime},C^{\prime})$ of the game $(\psi,P,C)$ such that
$\psi(P^{\prime},C^{\prime})=\mu.$

Next, we show that any stable matching rule implements the individually
rational correspondence in Nash equilibrium.666This result generalizes the
result first presented by Alcalde (1996) for the marriage market and then
extended by Sotomayor (2012) for the many-to-one matching market with
responsive preferences.

###### Theorem 1

Let $(P^{\prime},C^{\prime})$ be a market and let
$\psi:\mathcal{P}\times\mathcal{C}\rightarrow\mathcal{M}$ be a stable matching
rule. Then, the game $(\psi,P,C)$ implements the individually rational
correspondence $\mathcal{I}$ in Nash equilibrium.

_Proof._ Let $(P,C)$ be a market and $\psi$ a stable matching rule. In order
to show that the game $(\psi,P,C)$ implements $\mathcal{I}$ in Nash
equilibrium, we need to prove items (i) and (ii) of Definition 3.

  1. $\boldsymbol{(i)}$

Let $(P^{\prime},C^{\prime})$ be a Nash equilibrium of the game $(\psi,P,C)$.
We prove that $\psi(P^{\prime},C^{\prime})\in\mathcal{I}(P,C)$. Assume that
$\psi(P^{\prime},C^{\prime})$ is blocked by a worker $w$, $\emptyset
P_{w}\psi(P^{\prime},C^{\prime})(w)$. Then the worker can improve by choosing
the strategy in which no firm is acceptable i.e.,
$P^{\prime\prime}_{w}=\emptyset$. Thus,
$\psi(P^{\prime\prime}_{w},P^{\prime}_{-w},C^{\prime})(w)P_{w}\psi(P^{\prime},C^{\prime})(w)$
contradicting that $(P^{\prime},C^{\prime})$ is a Nash equilibrium of the
game. Hence, $\psi(P^{\prime},C^{\prime})$ is not blocked by any worker under
$(P,C)$. Now, we prove that $\psi(P^{\prime},C^{\prime})$ is not blocked by
any firm. Let $f\in F$ since $(P^{\prime},C^{\prime})$ is a Nash equilibrium,
for each strategy $C^{\prime\prime}_{f}$ we have that

$\psi(P^{\prime},C^{\prime})(f)=C_{f}\left(\psi(P^{\prime},C^{\prime})(f)\cup\psi(P^{\prime},C^{\prime\prime}_{f},C^{\prime}_{-f})(f)\right).$
(2)

Choose a choice function $C^{\prime\prime}_{f}$ such that
$C^{\prime\prime}_{f}(W)=\emptyset$. Since $\psi$ is a stable matching rule,
$\psi(P^{\prime},C^{\prime\prime}_{f},C^{\prime}_{-f})(f)=\emptyset$. Hence,
(2) becomes
$\psi(P^{\prime},C^{\prime})(f)=C_{f}\left(\psi(P^{\prime},C^{\prime})(f)\right)$
and $\psi(P^{\prime},C^{\prime})$ is not blocked by any firm under $(P,C)$.
Therefore, $\psi(P^{\prime},C^{\prime})\in\mathcal{I}(P,C)$.

  2. $\boldsymbol{(ii)}$

We need to prove that for each $\mu\in\mathcal{I}(P,C)$ there is a Nash
equilibrium $(P^{\star},C^{\star})$ of $(\psi,P,C)$ such that
$\psi(P^{\star},C^{\star})=\mu$. In order to do so, given
$\mu\in\mathcal{I}(P,C)$, we define $(P^{\star},C^{\star})$ in which each firm
$f$ declares $C^{\star}_{f}$ such that:

     * $\diamond$

for each $S\subseteq W$, $C^{\star}_{f}(S)=\mu(f)\cap S.$

     * $\diamond$

$C^{\star}_{f}(S^{\prime})\subsetneq C^{\star}_{f}(S)$ for each
$S^{\prime}\subsetneq S\subseteq\mu(f)$

and each worker $w$ declares $P_{w}^{\star}=\mu(w),\emptyset$. It can be
verified that $C^{\star}_{f}$ is substitutable for each firm $f\in F.$ Now, we
need to prove that (1) $\psi(P^{\star},C^{\star})=\mu$, and (2)
$(P^{\star},C^{\star})$ is a Nash equilibrium of the game.

    1. (1)

$\boldsymbol{\psi(P^{\star},C^{\star})=\mu}$. Since $\psi$ is a stable
matching rule under $(P^{\star},C^{\star})$, to prove (1) it is enough to show
that $S(P^{\star},C^{\star})=\\{\mu\\}$. First, we consider the DA algorithm
777See Roth and Sotomayor (1990) for more details. when workers make offers
under $(P^{\star},C^{\star})$. On the one hand, each unemployed worker under
$\mu$ makes no offers in any step of the DA algorithm, that is, it is an
unemployed worker under $\mu_{W}(P^{\star},C^{\star})$. On the other hand,
each worker $w$ assigned to a firm $f$ under $\mu$ makes an offer to the firm
$f$. Since $C^{\star}_{f}\left(\mu(f)\right)=\mu(f)$, firm $f$ accepts all
offers received. So $\mu_{W}(P^{\star},C^{\star})=\mu.$ Second, we consider
the DA algorithm when firms make offers under $(P^{\star},C^{\star})$. Notice
that each unmatched firm under $\mu$ makes no offers in any step of the DA
algorithm, i.e., it is an unmatched firm under $\mu_{F}(P^{\star},C^{\star})$.
For each matched firm under $\mu$, $C^{\star}_{f}(W)=\mu(f)$, so it makes
offers to each worker $w$ in $\mu(f)$. Since each worker $w$ in $\mu(f)$ has
only firm $f$ as acceptable, she accepts it. So
$\mu_{F}(P^{\star},C^{\star})=\mu.$ Therefore,
$\mu_{F}(P^{\star},C^{\star})=\mu_{W}(P^{\star},C^{\star})=\mu$, i.e. $\mu$ is
the unique stable matching under $(P^{\star},C^{\star})$.

    2. (2)

$\boldsymbol{(P^{\star},C^{\star})}$ is a Nash equilibrium of
$\boldsymbol{(\psi,P,C)}.$ Otherwise, there are two cases to consider:

Case (2.1): There is a firm $\boldsymbol{f\in F}$ and a choice function
$\boldsymbol{\widehat{C}_{f}}$ such that

$\boldsymbol{\psi(P^{\star},C^{\star})(f)\succeq_{f}\psi(P^{\star},\widehat{C}_{f},C^{\star}_{-f})(f)}$
does not hold. The strategy profile $(\widehat{C}_{f},C^{\star}_{-f})$ is
denoted by $\widehat{C}$. Thus, $\psi(P^{\star},C^{\star})(f)\neq
C_{f}\left(\psi(P^{\star},C^{\star})(f)\cup\psi(P^{\star},\widehat{C})(f)\right).$
Then, there is $w^{\prime}\in
C_{f}\left(\psi(P^{\star},C^{\star})(f)\cup\psi(P^{\star},\widehat{C})(f)\right)$
such that $w^{\prime}\notin\psi(P^{\star},C^{\star})(f)=\mu(f).$ Hence,
$w^{\prime}\in\psi(P^{\star},\widehat{C})(f)$ and $w^{\prime}\notin\mu(f).$
Thus, definition of $P^{\star}$ implies

$\mu(w^{\prime})P^{\star}_{w^{\prime}}\emptyset
P^{\star}_{w^{\prime}}f=\psi(P^{\star},\widehat{C})(w^{\prime}).$

Contradicting $\psi(P^{\star},\widehat{C})\in I(P^{\star},\widehat{C})$.

Case (2.2): There is a worker $\boldsymbol{w\in W}$ and a strategy
$\boldsymbol{\widehat{P}_{w}}$ such that
$\boldsymbol{\psi(P^{\star},C^{\star})(w)R_{w}\psi(\widehat{P}_{w},P^{\star}_{-w},C^{\star}_{f})(w)}$
does not hold. The strategy profile $(\widehat{P}_{w},P^{\star}_{-w})$ is
denoted by $\widehat{P}$. Thus,

$\psi(\widehat{P},C^{\star}_{f})(w)P_{w}\psi(P^{\star},C^{\star})(w).$ (3)

Let $f^{\prime}\in F$ such that
$f^{\prime}=\psi(\widehat{P},C^{\star}_{f})(w)$. By (1),
$\psi(P^{\star},C^{\star})(w)=\mu(w)$ and (3) becomes $f^{\prime}P_{w}\mu(w).$
The fact that $w\notin\mu(f^{\prime})$ and the definition of $C^{\star}$ imply
that
$C^{\star}_{f^{\prime}}\left(\psi(\widehat{P},C^{\star})(f^{\prime})\right)\neq\psi(\widehat{P},C^{\star})(f^{\prime})$
contradicting $\psi(\widehat{P},C^{\star})\in I(\widehat{P},C^{\star})$.

By Cases (2.1) and (2.2), $(P^{\star},C^{\star})$ is a Nash equilibrium of
$(\psi,P,C).$

Therefore by (i) and (ii), $(\psi,P,C)$ implements in Nash equilibrium the
individually rational correspondence $\mathcal{I}$. $\square$

The existence of Nash equilibria follows from Theorem 1. A natural question is
which is the game that implements stable matchings in Nash equilibrium. The
answer to this question is the focus of the next subsection.

###  3.2 The stable correspondence

Now consider a matching game in which the players are only the workers. Given
the profile of choice functions $C$, a social choice correspondence
$\Psi^{C}:\mathcal{P}\rightarrow 2^{\mathcal{M}}$ selects, for each strategy
profile $P^{\prime}\in\mathcal{P}$, a subset of matchings
$\Psi^{C}(P^{\prime})\subseteq\mathcal{M}.$ The stable correspondence
$\mathcal{S}^{C}:\mathcal{P}\rightarrow 2^{\mathcal{M}}$ selects for each
strategy profile $P^{\prime}\in\mathcal{P}$, the set of stable matchings
$S(P^{\prime},C)$. Given the profile of choice functions $C$, a (matching)
rule is a function $\psi^{C}:\mathcal{P}\rightarrow\mathcal{M}$ that selects
for each strategy profile $P^{\prime}\in\mathcal{P}$ a matching
$\psi^{C}(P^{\prime})\in\mathcal{M}.$ A rule $\psi^{C}$ is stable if
$\psi(P^{\prime})\in S(P^{\prime},C)$ for each
$(P^{\prime},C)\in\mathcal{P}\times\mathcal{C}.$

Given the profile of choice functions $C$, the firm-optimal stable rule
$\psi^{C}_{F}$ is a function $\psi_{F}^{C}:\mathcal{P}\rightarrow\mathcal{M}$
that selects for each strategy profile $P^{\prime}\in\mathcal{P}$ the firm-
optimal stable matching $\psi^{C}_{F}(P^{\prime})$ under $(P^{\prime},C).$

Given $P\in\mathcal{P}$ and the firm-optimal stable rule $\psi^{C}_{F}$ the
(matching) game induced by $P$ and $\psi^{C}_{F}$ is denoted by
$(\psi^{C}_{F},P).$ A strategy profile $P^{\prime}$ is a Nash equilibrium of
$(\psi^{C}_{F},P)$, if no worker can achieve a better outcome deviating from
her strategy, assuming that the other workers do not deviate from the strategy
profile $P^{\prime}$. Formally,

###### Definition 4

Let be $(\psi^{C}_{F},P)$ the game induced by $P$ and the stable rule
$\psi^{C}_{F}$. A strategy profile $P^{\prime}$ is a Nash equilibrium of
$(\psi^{C}_{F},P)$ if for each $w\in W$,
$\psi(P^{\prime},C^{\prime})(w)R_{w}\psi(\widehat{P}_{w},P^{\prime}_{-w},C^{\prime})(w)$
for each strategy $\widehat{P}_{w}$ of $w$.

###### Definition 5

We say that the game $(\psi^{C}_{F},P)$ implements the social choice
correspondence $\Psi^{C}$ in Nash equilibrium if,

  1. (i)

for each Nash equilibrium $P^{\prime}$ of $(\psi^{C}_{F},P)$,
$\psi^{C}_{F}(P^{\prime})\in\Psi^{C}(P)$,

  2. (ii)

for each matching $\mu\in\Psi^{C}(P)$ there is a Nash equilibrium $P^{\prime}$
of the game $(\psi^{C}_{F},P)$ such that $\psi^{C}_{F}(P^{\prime})=\mu.$

By means of an additional condition on firms’ choice functions, we can
implement the stable correspondence in Nash equilibrium. This additional
condition is the “law of aggregate demand”, which says that when a firm
chooses from an expanded set, it hires at least as many workers as before.
Formally,

###### Definition 6

Choice function $C_{f}$ satisfies  law of aggregate demand (LAD) if
$S^{\prime}\subseteq S\subseteq W$ implies
$|C_{f}(S^{\prime})|\leq|C_{f}(S)|.$

We denote by $\mathcal{C}_{LAD}$ the set of all firms’ choice functions
$C\in\mathcal{C}$ such that $C_{f}$ satisfies LAD for each $f\in F$. The
following theorem asserts that the firm-optimal stable rule implements, in
Nash equilibrium, the stable correspondence.

###### Theorem 2

Let $(P,C)$ be a market, $C\in\mathcal{C}_{LAD},$ and let $\psi^{C}_{F}$ be
the firm-optimal stable matching rule. Then, the game $(\psi^{C}_{F},P)$
implements the stable correspondence $\mathcal{S}^{C}$ in Nash equilibrium.

_Proof._ Let $P$ be a preference profile, $C$ be a profile of choice functions
and $\psi^{C}_{F}$ the firm-optimal stable rule. In order to show that the
game $(\psi^{C}_{F},P)$ implements $\mathcal{S}^{C}$ in Nash equilibrium, we
need to prove items (i) and (ii) of Definition 5.

  1. $\boldsymbol{(i)}$

Let $P^{\prime}$ be a Nash equilibrium of the game $(\psi^{C}_{F},P)$. We
prove that $\psi^{C}_{F}(P^{\prime})\in\mathcal{S}^{C}(P).$ An argument
analogous to the proof of Theorem 1 (i), shows that $\psi^{C}_{F}$ is not
blocked by any worker. Since $\psi^{C}_{F}$ is a stable rule under $(P,C)$,
$\psi^{C}_{F}$ is not blocked by any firm. Therefore
$\psi^{C}_{F}(P^{\prime})\in I(P,C).$ Assume that
$\psi^{C}_{F}(P^{\prime})\notin S(P,C)$. Thus, there is a blocking pair
$(f,w)$ for $\psi^{C}_{F}(P^{\prime}),$

$w\in
C_{f}\left(\psi^{C}_{F}(P^{\prime})(f)\cup\left\\{w\right\\}\right)~{}\text{and}~{}fP_{w}\psi^{C}_{F}(P^{\prime})(w).$
(4)

Now, consider the strategy profile
$P^{\prime\prime}=(P^{\prime\prime}_{w},P^{\prime}_{-w})$ where
$P^{\prime\prime}_{w}=f,\emptyset$ . We claim that
$\psi^{C}_{F}(P^{\prime\prime})(w)=f$. Otherwise, since
$\psi^{C}_{F}(P^{\prime\prime})\in I(P^{\prime\prime},C)$,
$\psi^{C}_{F}(P^{\prime\prime})(w)=\emptyset$. The definition of
$P^{\prime\prime}$ implies that
$fP^{\prime\prime}_{w}\psi^{C}_{F}(P^{\prime\prime})(w)=\emptyset.$ If $w\in
C_{f}\left(\psi^{C}_{F}(P^{\prime\prime})(f)\cup\left\\{w\right\\}\right),$
then $(f,w)$ blocks $\psi^{C}_{F}(P^{\prime\prime})$ under
$(P^{\prime\prime},C)$, contradicting that $\psi^{C}_{F}(P^{\prime\prime})\in
S(P^{\prime\prime},C)$. Thus, $w\notin
C_{f}\left(\psi^{C}_{F}(P^{\prime\prime})(f)\cup\left\\{w\right\\}\right).$
Then, by substitutability,
$w\notin
C_{f}\left(\psi^{C}_{F}(P^{\prime\prime})(f)\cup\psi^{C}_{F}(P^{\prime})(f)\right)$.
Applying Theorem 1 in Crawford (1981),888The version of this theorem for a
many-to-one matching market, where all firms have substitutable choice
functions, states that by removing one or more workers from the market, the
firm-optimal stable matching rule in the new reduced market is weakly worse
for all firms than the firm-optimal stable matching rule in the original
matching market. In this case, we consider the reduced market
$(F,W\setminus\\{w\\},P^{\prime}_{-w})$.

$w\notin
C_{f}\left(\psi^{C}_{F}(P^{\prime\prime})(f)\cup\psi^{C}_{F}(P^{\prime})(f)\right)=\psi^{C}_{F}(P^{\prime})(f).$
(5)

Now, using (1) and (5) we get

$w\notin
C_{f}\left(\psi^{C}_{F}(P^{\prime\prime})(f)\cup\psi^{C}_{F}(P^{\prime})(f)\right)=C_{f}\left(C_{f}\left(\psi^{C}_{F}(P^{\prime\prime})(f)\cup\psi^{C}_{F}(P^{\prime})(f)\right)\cup\left\\{w\right\\}\right)$
$=C_{f}\left(\psi^{C}_{F}(P^{\prime})(f)\cup\left\\{w\right\\}\right).$

Thus, $w\notin
C_{f}\left(\psi^{C}_{F}(P^{\prime})(f)\cup\left\\{w\right\\}\right)$, but this
contradicts (4) and the claim is proven. Therefore,
$\psi^{C}_{F}(P^{\prime\prime})(w)=f.$ By (4), we have
$\psi^{C}_{F}(P^{\prime\prime})(w)P_{w}\psi^{C}_{F}(P^{\prime})(w)$, then
$P^{\prime}$ is not a Nash equilibrium, contradicting our hypothesis.
Therefore $\psi^{C}_{F}(P^{\prime})\in\mathcal{S}^{C}(P).$

  2. $\boldsymbol{(ii)}$

We need to prove that for each $\mu\in\mathcal{S}^{C}(P)$ there is a Nash
equilibrium $P^{\star}$ of $(\psi^{C}_{F},P)$ such that
$\psi^{C}_{F}(P^{\star})=\mu$. In order to do so, given
$\mu\in\mathcal{S}^{C}(P)$ we define $P^{\star}$ in which each worker $w$
declares $P^{\star}_{w}=\mu(w),\emptyset$. Now, we need to prove that (1)
$\psi^{C}_{F}(P^{\star})=\mu$, and (2) $P^{\star}$ is a Nash equilibrium of
the game.

    1. (1)

$\boldsymbol{\psi^{C}_{F}(P^{\star})=\mu}$. Since $\psi^{C}_{F}$ is a stable
matching rule under $(P^{\star},C)$, to prove (1) is suffices to show that
$S(P^{\star},C)=\\{\mu\\}$.

Following similar reasoning to the ones in the proof of Theorem 1, the item
(ii) part (1), $\mu_{W}(P^{\star})=\mu$. Assume that
$\mu_{W}(P^{\star})\neq\mu_{F}(P^{\star})$, then there is $w\in W$ such that
$\mu_{W}(P^{\star})(w)\neq\mu_{F}(P^{\star})(w).$ Since $\mu(w)$ is the only
acceptable firm under $P^{\star}$, $\mu_{F}(P^{\star})(w)=\emptyset$, but this
contradicts the Rural Hospital Theorem 999The Rural Hospital Theorem is proven
in different contexts by many authors (see McVitie and Wilson, 1971; Roth,
1984, 1985; Martínez et al., 2000; Alkan, 2002; Kojima, 2012, among others).
The version of this theorem for a many-to-many matching market where all
agents have substitutable choice functions satisfying _LAD_ , that also
applies in our setting, is presented in Alkan (2002) and states that each
agent is matched with the same number of partners in every stable matching..
Therefore $\mu_{W}(P^{\star})=\mu_{F}(P^{\star})=\mu,$ i.e. $\mu$ is the
unique stable matching under $(P^{\star},C)$.

    2. (2)

$\boldsymbol{P^{\star}}$ is a Nash equilibrium of
$\boldsymbol{(\psi^{C}_{F},P).}$ Otherwise, there is a worker
$\boldsymbol{w\in W}$ and a strategy $\boldsymbol{\widehat{P}_{w}}$ such that
$\boldsymbol{\psi^{C}_{F}(P^{\star})(w)R_{w}\psi_{F}(\widehat{P}_{w},P^{\star}_{-w})(w)}$
does not hold. The strategy profile $(\widehat{P}_{w},P^{\star}_{-w})$ is
denoted by $\widehat{P}.$ Thus,

$\psi^{C}_{F}(\widehat{P})(w)P_{w}\psi^{C}_{F}(P^{\star})(w).$ (6)

Since $\psi^{C}_{F}(P^{\star})(w)=\mu(w)$ and $\mu\in I(P,C)$, there is
$f^{\prime}\in F$ such that $f^{\prime}=\psi^{C}_{F}(\widehat{P})(w)$. If
$w\in
C_{f^{\prime}}\left(\psi^{C}_{F}(P^{\star})(f^{\prime})\cup\\{w\\}\right)$,
this together with (6) imply that $(f^{\prime},w)$ blocks $\mu$ under $(P,C)$.
This is a contradiction. Therefore,

$w\notin
C_{f^{\prime}}\left(\psi^{C}_{F}(P^{\star})(f^{\prime})\cup\\{w\\}\right).$
(7)

We claim that

$\psi^{C}_{F}(\widehat{P})(f^{\prime})\subsetneq\psi^{C}_{F}(P^{\star})(f^{\prime})\cup\\{w\\}.$
(8)

Let $\widehat{w}\neq w$ such that
$\widehat{w}\in\psi^{C}_{F}(\widehat{P})(f^{\prime})$. Definition of
$\widehat{P}$ and $\psi^{C}_{F}(\widehat{P})\in I(\widehat{P})$ imply that
$f^{\prime}=\psi^{C}_{F}(\widehat{P})(\widehat{w})=\mu(\widehat{w})=\psi^{C}_{F}(P^{\star})(\widehat{w}).$
Thus, $\widehat{w}\in\psi^{C}_{F}(P^{\star})(f^{\prime}).$ Now, assume that

$\psi^{C}_{F}(\widehat{P})(f^{\prime})=\psi^{C}_{F}(P^{\star})(f^{\prime})\cup\\{w\\}.$
(9)

Using, $\psi^{C}_{F}(\widehat{P})\in I(\widehat{P},C)$, definition of
$\widehat{P}$ and (9), it follows that

$\psi^{C}_{F}(\widehat{P})(f^{\prime})=C_{f^{\prime}}\left(\psi^{C}_{F}(\widehat{P})(f^{\prime})\right)=C_{f^{\prime}}\left(\psi^{C}_{F}(P^{{\star}})(f^{\prime})\cup\\{w\\}\right).$

Using, (7) and $\psi^{C}_{F}(P^{\star})\in I(P^{\star},C)$, it follows that

$C_{f^{\prime}}\left(\psi^{C}_{F}(P^{\star})(f^{\prime})\cup\\{w\\}\right)=C_{f^{\prime}}\left(\psi^{C}_{F}(P^{\star})(f^{\prime})\right)=\psi^{C}_{F}(P^{\star})(f^{\prime}).$

Hence,
$\psi^{C}_{F}(\widehat{P})(f^{\prime})=\psi^{C}_{F}(P^{\star})(f^{\prime})$
contradicting (6). This completes the proof of the Claim. Therefore, there is
$w^{\star}\in\psi^{C}_{F}(P^{\star})(f^{\prime})\setminus\psi^{C}_{F}(\widehat{P})(f^{\prime})$.
If $w^{\star}\notin
C_{f^{\prime}}\left(\psi^{C}_{F}(\widehat{P})(f^{\prime})\cup\\{w^{\star}\\}\right)$.
Using (8), substitutability and definition of $\widehat{P}$ , $w^{\star}\notin
C_{f^{\prime}}\left(\psi^{C}_{F}(P^{\star})(f^{\prime})\cup\\{w\\}\right)$.
Using (7) and $\psi^{C}_{F}(P^{\star})\in I(P^{\star},C)$ we have that,

$w^{\star}\notin
C_{f^{\prime}}\left(\psi^{C}_{F}(P^{\star})(f^{\prime})\cup\\{w\\}\right)=C_{f^{\prime}}\left(\psi^{C}_{F}(P^{\star})(f^{\prime})\right)=\psi^{C}_{F}(P^{\star})(f^{\prime}).$

Thus, $w^{\star}\notin\psi^{C}_{F}(P^{\star})(f^{\prime})$ contradicting that
$w^{\star}\in\psi^{C}_{F}(P^{\star})(f^{\prime})\setminus\psi^{C}_{F}(\widehat{P})(f^{\prime})$.
Then,

$w^{\star}\in
C_{f^{\prime}}\left(\psi^{C}_{F}(\widehat{P})(f^{\prime})\cup\\{w^{\star}\\}\right).$
(10)

The fact that
$\mu(w^{\star})=\psi^{C}_{F}(P^{\star})(w^{\star})\neq\psi^{C}_{F}(\widehat{P})(w^{\star})$
and the definition of $\widehat{P}$ imply that,
$f^{\prime}=\mu(w^{\star})P^{\star}_{w^{\star}}\psi^{C}_{F}(\widehat{P})(w^{\star})$.
This together with (10) imply that $(f^{\prime},w^{\star})$ blocks
$\psi^{C}_{F}(\widehat{P})$ under $(\widehat{P},C)$, contradicting the
stability of $\psi^{C}_{F}(\widehat{P})$ under $(\widehat{P},C)$.

Therefore, $P^{\star}$ is a Nash equilibrium.

Finally, by (i) and (ii), $(\psi^{C}_{F},P)$ implements in Nash equilibrium
the stable correspondence $\mathcal{S}^{C}$. $\square$

##  4 Concluding Remarks

The main motivation of this paper is to provide a framework to study the Nash
equilibrium solutions of the game induced by stable rules. In a many-to-one
matching market with substitutable choice functions, we show that any stable
matching rule implements, in Nash equilibrium, the individually rational
matchings. Moreover, when only workers play strategically and firms’ choice
functions satisfy the law of aggregate demand, we show that the firm-optimal
stable rule $\psi^{C}_{F}$ implements the stable correspondence in Nash
equilibrium. The analogous result with workers telling the truth and firms
acting strategically does not hold. That is, if we consider a game in which
the players are only the firms, we cannot implement the stable correspondence
in Nash equilibrium. This fact was already noted by Sotomayor (2012), even
under a more restrictive model with responsive preferences.

It is usual in the literature to study many-to-one markets assuming that
firms’ preferences are responsive. This is due to the close relationship
between this market with responsive preferences and the marriage market , (see
Roth and Sotomayor, 1990, for a thorough survey on this fact) . However, when
firms are endowed with substitutable preferences (a much less restrictive
requirement), this relation with the marriage market no longer holds. Thus,
extending the results of a many-to-one market with responsive preferences to
substitutable choice functions is not straightforward.

The study of the implementability of several solution concepts under other
equilibria notions is an interesting topic for future research.

## References

  * Alcalde (1996) Alcalde, J. (1996): “Implementation of stable solutions to marriage problems,” _Journal of Economic Theory_ , 69, 240–254.
  * Alcalde and Romero Medina (2000) Alcalde, J. and A. Romero Medina (2000): “Simple Mechanisms to Implement the Core of College Admissions Problems,” _Games and Economic Behavior_ , 31, 294–302.
  * Alkan (2002) Alkan, A. (2002): “A class of multipartner matching markets with a strong lattice structure,” _Economic Theory_ , 19, 737–746.
  * Blair (1988) Blair, C. (1988): “The lattice structure of the set of stable matchings with multiple partners,” _Mathematics of Operations Research_ , 13, 619–628.
  * Crawford (1981) Crawford, V. P. (1991): “Comparative statics in matching markets,” _Journal of Economic Theory_ , 54, 389–400.
  * Dubins and Freedman (1981) Dubins L. E. and D. A. Freedman (1981): “Machiavelli and the Gale-Shapley algorithm,” _The American Mathematical Monthly_ , 88, 485–494.
  * Ehlers (2004) Ehlers L.  (2004): “Monotonic and implementable solutions in generalized matching problems,” _Journal of Economic Theory_ , 114, 358–369.
  * Gale and Shapley (1962) Gale, D. and L. Shapley (1962): “College admissions and the stability of marriage,” _The American Mathematical Monthly_ , 69, 9–15.
  * Haake and Klaus (2009) Haake, C. J. and B. Klaus (2009): “Monotonicity and Nash implementation in matching markets with contracts,” _Economic Theory_ , 41, 393–410.
  * Hatfield and Milgrom (2005) Hatfield, J. and P. Milgrom (2005): “Matching with contracts,” _American Economic Review_ , 95, 913–935.
  * Kara and Sönmez (1996) Kara, T. and T. Sönmez (1996): “Nash implementation of matching rules,” _Journal of Economic Theory_ , 68, 425–439.
  * Kara and Sönmez (1997) Kara, T. and T. Sönmez (1997): “Implementation of college admission rules,” _Economic Theory_ , 9, 197–218.
  * Kelso and Crawford (1982) Kelso, A., and V. Crawford (1982): “Job matching, coalition formation, and gross substitutes,” _Econometrica: Journal of the Econometric Society_ , 50, 1483–1504.
  * Kojima (2012) Kojima, F. (2012): “The "rural hospital theorem" revisited,” _International Journal of Economic Theory_ , 8, 67–76.
  * Manasero and Oviedo (2022) Manasero, P., and J. Oviedo (2022): “General manipulability theorem for a matching model” _arXiv preprint arXiv:2210.06549_.
  * Martínez et al. (2000) Martínez, R., J. Massó, A. Neme, and J. Oviedo (2000): “Single agents and the set of many-to-one stable matchings,” _Journal of Economic Theory_ , 91, 91–105.
  * Martínez et al. (2004) Martínez, R., J. Massó, A. Neme, and J. Oviedo (2004): “On group strategy-proof mechanisms for a many-to-one matching model,” _International Journal of Game Theory_ , 33, 115–128.
  * Maskin (1977) Maskin, E. (1977): “Nash equilibrium and welfare optimality,” _mimeo, MIT_.
  * Maskin (1999) Maskin, E. (1999): “Nash equilibrium and welfare optimality,” _Review of Economic Studies_ , 66, 23–38.
  * McVitie and Wilson (1971) McVitie, D. and L. Wilson (1971): “The stable marriage problem,” _Communications of the ACM_ , 14, 486–490.
  * Moore and Repullo (1990) Moore, J. and R. Repullo (1990): “Nash implementation: a full characterization,” _Econometrica_ , 58, 1083–1099.
  * Roth (1982) Roth, A. (1982): “The economics of matching: stability and incentives,” _Mathematics of Operations Research_ , 7, 617–628.
  * Roth (1984) Roth, A. (1984): “The evolution of the labor market for medical interns and residents: a case study in game theory,” _Journal of Political Economy_ , 92, 991–1016.
  * Roth (1985) Roth, A. (1985): “The college admissions problem is not equivalent to the marriage problem,” _Journal of Economic Theory_ , 36, 277–288.
  * Roth and Sotomayor (1990) Roth, A. and M. Sotomayor (1990): _Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis_ , Cambridge University Press.
  * Sotomayor (1996) Sotomayor, M. (1996): “Admission mechanisms of students to colleges,” _A game-theoretic modeling and analysis. Braz Rev Econ_ , 16, 25–63.
  * Sotomayor (1999) Sotomayor, M. (1999): “Three remarks on the many-to-many stable matching problem,” _Mathematical Social Sciences_ , 38, 55–70.
  * Sotomayor (2008) Sotomayor, M. (2008): “The stability of the equilibrium outcomes in the admission games induced by stable matching rules,” _International Journal of Game Theory_ , 36, 621–640.
  * Sotomayor (2012) Sotomayor, M. (2012): “A further note on the college admission game,” _International Journal of Game Theory_ , 41, 179–193.
  * Yamato (1992) Yamato, T. (1992): “On Nash implementation of social choice correspondences,” _Games and Economic Behavior_ , 4, 484–492.


# Testing Symmetry for Bivariate Copulas using Bernstein Polynomials

Guanjie Lyu
Department of Mathematics and Statistics  University of Windsor  Canada
M. Belalia
Department of Mathematics and Statistics  Corresponding author:
Mohamed.Belalia @ uwindsor.ca University of Windsor  Canada

###### Abstract

In this work, tests of symmetry for bivariate copulas are introduced and
studied using empirical Bernstein copula process. Three statistics are
proposed and their asymptotic properties are established. Besides, a
multiplier bootstrap Bernstein version is investigated for implementation
purpose. Simulation study and real data application showed that the Bernstein
tests outperform the tests based on the empirical copula.

Keywords: Testing symmetry; Empirical Bernstein copula; Multiplier bootstrap;
Empirical copula process.

##  1 Introduction

Asymmetric copulas have massive explorations and applications in recent years.
They are a powerful tools to capture the asymmetric dependence structure and
has been applied in many fields, for instance, (Grimaldi and Serinaldi, 2006,
flood frequency analysis), (Wu, 2014, reliability modeling) and (Zhang et al.,
2018, ocean data modeling). Meanwhile, more attention is given to testing and
identifying the symmetric nature of a copula. Genest et al. (2012) proposed
tests based on empirical copula, Bahraoui et al. (2018) used empirical copula
characteristic function to construct the test, Jaser and Min (2020) developed
a test by representing copula as a mixture of two conditional distribution
functions, Beare and Seo (2020) studied a randomization procedure and so on.
For $d$-variate symmetry tests, the work of Genest et al. (2012) was extended
by Harder and Stadtmüller (2017), Bahraoui and Quessy (2022) investigated
tests based on Lévy measures.

In this work, tests of symmetry for bivariate copulas based on empirical
Bernstein copula process are proposed. Specifically, consider a random pair
$(X,Y)$ with cumulative distribution function $F_{XY}(x,y)=\mathbb{P}(X\leq
x,Y\leq y)$ and continuous margins $F_{X}$ and $F_{Y}$. According to Sklar
(1959), there exists a unique copula function $C$ such that

$F_{XY}(x,y)=C\\{F_{X}(x),G_{Y}(y)\\}.$

To detect the symmetry of copula $C$, one would like to test the following
hypotheses

${}\begin{cases}\mathscr{H}_{0}\mathrel{\mathop{\ordinarycolon}}C(u,v)=C(v,u)&\forall(u,v)\in[0,1]^{2}\\\
&\text{versus}\\\ \mathscr{H}_{1}\mathrel{\mathop{\ordinarycolon}}C(u,v)\neq
C(v,u)&\exists(u,v)\in[0,1]^{2}.\end{cases}$ (1)

The symmetry property of bivariate copulas has intimate relation with the
symmetry of corresponding random pairs and was discussed in Nelsen (1993,
2006, 2007). It was shown that $X$ and $Y$ are exchangeable, $i.e.$,
$F_{XY}(x,y)=F_{XY}(y,x)$ if an only if $F_{X}=F_{Y}$ and $C(u,v)=C(v,u)$ for
$(u,v)\in[0,1]^{2}$. Specifically, for identically distributed margins, the
symmetry structure of the copula can determine the exchangeability of the
random variables $X$ and $Y$. Identifying the equality of margins is well-
developed and can be investigated using Kolmogorov–Smirnov test or Cramér–von
Mises test. For a positively dependent survival data setting, see Fujii (1989)
and for a high-dimensional data setting, see Cousido-Rocha et al. (2019). As
the equality of margins was already discussed vigorously by many others, the
contribution of detecting the symmetry of copula turns more attracting for
deciding the exchangability of two random variables. Moreover, before fitting
a specific copula model to the data, identifying the symmetric structure can
assist us to choose an appropriate model, for example, Archimedean copulas are
symmetric.

Empirical Bernstein copula has drawn attention in recent years due to free
boundary bias properties and tractability of implementation. Indeed, one only
needs to select an appropriate integer parameter $m$ during deployment.
Theoretical properties of this estimator have been well-studied, see Sancetta
and Satchell (2004), Janssen et al. (2012), Belalia et al. (2017) and Segers
et al. (2017), among others. However, computational investigation of the
empirical processes based on this estimator are barely unexplored. To fill
this gap, we proposed a smooth version of multiplier bootstrap method for
empirical Bernstein copula process with promising performance.

The rest of this paper is structured as follows. In Section 2, new test
statistics are presented and their asymptotic behaviours are examined. Section
3 develops the empirical Bernstein copula process multiplier bootstrap ans its
large sample behaviour . Simulation studies are carried out in Section 4. The
Ocean data is analyzed using the proposed methodology in Section 5. Some
concluding remarks are given in Section 6. Finally, the proofs are relegated
in the Appendix.

##  2 New testing procedures based on empirical Bernstein copula

###  2.1 Descriptive of the test statistics

Let $(X_{1},Y_{1}),\ldots,(X_{n},Y_{n})$ be a random sample from a bivariate
distribution $F_{XY}$ with continuous margins $F_{X}$ and $F_{Y}$. Also, let
$C$ be their associated copula. Unfortunately, this function is generally
unknown, hence it has to be estimated. The empirical copula introduced by
Rüschendorf (1976) is a natural nonparametric estimator of $C$ and is given by

$\widehat{C}_{n}(u,v)=\frac{1}{n}\sum_{i=1}^{n}{\mathbb{I}}\left(\widehat{U}_{i}\leq
u,\widehat{V}_{i}\leq v\right),$ (2)

where $\widehat{U}_{i}=n^{-1}\sum_{j=1}^{n}{\mathbb{I}}\left(X_{j}\leq
X_{i}\right)$ and
$\widehat{V}_{i}=n^{-1}\sum_{j=1}^{n}{\mathbb{I}}\left(Y_{j}\leq Y_{i}\right)$
are the empirical distributions of the margins.

The empirical copula is widely used for construction of nonparametric tests in
the literature, such as, test of independence, goodness-of-fit test among
others. However, it is not a continuous estimator for $C$, which mismatches
the continuity of copula function. To overcome this drawback, smoothed
empirical copulas were developed, for example, Morettin et al. (2010)
introduced wavelet-smoothed empirical copula for time series data, Gijbels and
Mielniczuk (1990), Fermanian et al. (2004), Chen and Huang (2007), Omelka et
al. (2009) considered kernel-smoothed empirical copula and Genest et al.
(2017) proposed empirical checkerboard copula. Here, the empirical Bernstein
copula is applied. This method has more appealing boundary properties
comparing with kernel method. Besides, the support of bivariate copula is
$[0,1]^{2}$ which meets the Bernstein polynomials assumption perfectly.

The empirical Bernstein copula estimator of order $m$ is defined as

$\displaystyle\widehat{C}_{n,m}(u,v)$
$\displaystyle=\sum_{k=0}^{m}\sum_{\ell=0}^{m}\widehat{C}_{n}\left(k/m,\ell/m\right)P_{m,k}(u)P_{m,\ell}(v)$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\sum_{k=0}^{m}\sum_{\ell=0}^{m}{\mathbb{I}}\left(\widehat{U}_{i}\leq
k/m,\widehat{V}_{i}\leq\ell/m\right)P_{m,k}(u)P_{m,\ell}(v),$ (3)

where $P_{m,k}(u)=\binom{m}{k}u^{k}(1-u)^{m-k}$ is the binomial probability
mass function. Note that, $m$ is dependent on $n$, in particular, if the
degree of Bernstein polynomials $m$ is equal to the sample size $n$, the
estimator (2.1) turns out to be the empirical beta copula developed by Segers
et al. (2017). Some bootstrap methods for empirical beta copula are
investigated in Kiriliouk et al. (2021) and see Kojadinovic and Yi (2022);
Kojadinovic (2022) for more recent studies. Based on the empirical Bernstein
copula, three statistics are proposed to test the symmetry as follows,

$\displaystyle R_{n,m}$
$\displaystyle=\int_{0}^{1}\int_{0}^{1}\Big{\\{}\widehat{C}_{n,m}(u,v)-\widehat{C}_{n,m}(v,u)\Big{\\}}^{2}\operatorname{d\\!}u\operatorname{d\\!}v,$
$\displaystyle S_{n,m}$
$\displaystyle=\int_{0}^{1}\int_{0}^{1}\Big{\\{}\widehat{C}_{n,m}(u,v)-\widehat{C}_{n,m}(v,u)\Big{\\}}^{2}\operatorname{d\\!}\widehat{C}_{n}(u,v),$
$\displaystyle T_{n,m}$
$\displaystyle=\sup_{(u,v)\in[0,1]^{2}}\left|\widehat{C}_{n,m}(u,v)-\widehat{C}_{n,m}(v,u)\right|.$
(4)

In what follows, the asymptotic behaviour of the three test statistics will be
studied, which include the asymptotic limit under both the null and
alternative hypotheses.

###  2.2 Asymptotic behaviour of the test statistics

As it will be seen the limits of the proposed test statistics are functional
of the unknown underlying copula $C$, therefore, some common assumptions are
needed before going further.

######  Assumption 1.

Assume that first-order partial derivatives $\dot{C}_{1}(u,v)=\partial
C(u,v)/\partial u,\dot{C}_{2}(u,v)=\partial C(u,v)/\partial v$ exist and are
continuous, respectively, on the sets $(0,1)\times[0,1]$ and
$[0,1]\times(0,1)$.

######  Assumption 2.

Assume that second-order partial derivatives
$\ddot{C}_{11}(u,v)=\partial^{2}C(u,v)/\partial
u^{2},\ddot{C}_{21}(u,v)=\partial^{2}C(u,v)/\partial u\partial v$ and
$\ddot{C}_{22}(u,v)=\partial^{2}C(u,v)/\partial v^{2}$ exist and are
continuous, respectively, on the sets $(0,1)\times[0,1],(0,1)\times(0,1)$ and
$[0,1]\times(0,1)$, and there exists a constant $K>0$ such that

$\left|\ddot{C}_{11}(u,v)\right|\leq\frac{K}{u(1-u)},\quad\left|\ddot{C}_{21}(u,v)\right|\leq
K\min\left(\frac{1}{u(1-u)},\frac{1}{v(1-v)}\right),\quad\left|\ddot{C}_{22}(u,v)\right|\leq\frac{K}{v(1-v)}.$

Note that, according to (Segers, 2012, Lemma 4.3), under Assumption 1-2,
$\dot{C}_{1}(u,v)$ and $\dot{C}_{2}(u,v)$ are Lipschitz continuous on the sets
$(0,1)\times[0,1]$ and $[0,1]\times(0,1)$, respectively.

Let $\mathbb{B}_{n,m},\mathbb{S}_{n,m}$ denote the empirical Bernstein copula
process and symmetrised empirical Bernstein copula process, namely, for all
$(u,v)\in[0,1]^{2}$.

$\mathbb{B}_{n,m}(u,v)=\sqrt{n}\Big{\\{}\widehat{C}_{n,m}(u,v)-C(u,v)\Big{\\}},\quad\mathbb{S}_{n,m}(u,v)=\sqrt{n}\Big{\\{}\widehat{C}_{n,m}(u,v)-\widehat{C}_{n,m}(v,u)\Big{\\}}.$

From the work of Segers et al. (2017), suppose $C$ satisfies Assumption 1 and
$n/m\to c>0$, then, in $\ell^{\infty}([0,1]^{2})$ (the Banach space of all
real-valued, bounded functions on $[0,1]^{2}$, equipped with sup-norm),

$\mathbb{B}_{n,m}(u,v)\rightsquigarrow\mathbb{B}_{C}(u,v)=\mathbb{C}(u,v)-\dot{C}_{1}(u,v)\mathbb{C}(u,1)-\dot{C}_{2}(u,v)\mathbb{C}(1,v),\quad\text{as
$n\rightarrow\infty$},$ (5)

where $\mathbb{C}$ is a $C$-Brownian bridge whose covariance function is

$\Gamma_{\mathbb{C}}(u,v,s,t)=C(\min(u,s),\min(v,t))-C(u,v)C(s,t),$

for $(u,v),(s,t)\in[0,1]^{2}$ and $``\rightsquigarrow$” stands for the weak
convergence. Based on this result, the weak convergence of the symmetrised
empirical Bernstein copula process is presented in the following theorem.

######  Theorem 1.

Let $C$ be a symmetric copula satisfying Assumption 1, in addition, if $n/m\to
c>0$, then as $n\rightarrow\infty$, $\mathbb{S}_{n,m}$ converges weakly to a
Gaussian process $\mathbb{S}_{C}$ defined by

$\mathbb{S}_{C}(u,v)=\mathbb{D}(u,v)-\dot{C}_{1}(u,v)\mathbb{D}(u,1)-\dot{C}_{2}(u,v)\mathbb{D}(1,v),$

for all $(u,v)\in[0,1]^{2}$, where $\mathbb{D}$ is a $C$-Brownian bridge with
covariance function given at each $(u,v),(s,t)\in[0,1]^{2}$ by
$\Gamma_{\mathbb{D}}(u,v,s,t)=2\\{\Gamma_{\mathbb{C}}(u,v,s,t)-\Gamma_{\mathbb{C}}(u,v,t,s)\\}$.

###### Proof.

Under the null hypothesis, one can write that,

$\displaystyle\mathbb{S}_{n,m}(u,v)$
$\displaystyle=\mathbb{B}_{n,m}(u,v)+\sqrt{n}C(u,v)-\mathbb{B}_{n,m}(v,u)-\sqrt{n}C(v,u)$
$\displaystyle=\mathbb{B}_{n,m}(u,v)-\mathbb{B}_{n,m}(v,u).$

Further, it was shown in Segers et al. (2017), that
$\mathbb{B}_{n,m}\rightsquigarrow\mathbb{B}_{C}$ as $n\rightarrow\infty$.
Then, the desired result can be proven using continuous mapping theorem,
namely,

$\mathbb{S}_{n,m}\rightsquigarrow\mathbb{S}_{C},$

where $\mathbb{S}_{C}(u,v)=\mathbb{B}_{C}(u,v)-\mathbb{B}_{C}(v,u)$. ∎

Based on Theorem 1 and the fact that the proposed test statistics in (2.1) are
functional of $\mathbb{S}_{n,m}$, their asymptotic behaviours under the null
hypothesis can be established as follows.

######  Theorem 2.

Let $C$ be a symmetric copula satisfying Assumption 1, in addition, if $n/m\to
c>0$, then as $n\rightarrow\infty$,

$\displaystyle
nR_{n,m}=\int_{0}^{1}\int_{0}^{1}\big{\\{}\mathbb{S}_{n,m}(u,v)\big{\\}}^{2}\operatorname{d\\!}u\operatorname{d\\!}v$
$\displaystyle\rightsquigarrow\mathbb{L}_{R}=\int_{0}^{1}\int_{0}^{1}\big{\\{}\mathbb{S}_{C}(u,v)\big{\\}}^{2}\operatorname{d\\!}u\operatorname{d\\!}v$
$\displaystyle
nS_{n,m}=\int_{0}^{1}\int_{0}^{1}\big{\\{}\mathbb{S}_{n,m}(u,v)\big{\\}}^{2}\operatorname{d\\!}\widehat{C}_{n}(u,v)$
$\displaystyle\rightsquigarrow\mathbb{L}_{S}=\int_{0}^{1}\int_{0}^{1}\big{\\{}\mathbb{S}_{C}(u,v)\big{\\}}^{2}\operatorname{d\\!}C(u,v),$
$\displaystyle
n^{1/2}T_{n,m}=\underset{(u,v)\in[0,1]^{2}}{\sup}|\mathbb{S}_{n,m}(u,v)|$
$\displaystyle\rightsquigarrow\mathbb{L}_{T}=\underset{(u,v)\in[0,1]^{2}}{\sup}\left|\mathbb{S}_{C}(u,v)\right|.$

###### Proof of Theorem 2.

The proof is postponed to the appendix. ∎

More generally, the asymptotic behaviours of the test statistics under the
alternative hypothesis are given as follows.

######  Theorem 3.

Let $C$ be a copula satisfying Assumption 1, if $n/m\to c>0$, then as
$n\rightarrow\infty$,

$\displaystyle R_{n,m}$
$\displaystyle\xrightarrow{a.s.}R_{C}=\int_{0}^{1}\int_{0}^{1}\Big{\\{}C(u,v)-C(v,u)\Big{\\}}^{2}\operatorname{d\\!}u\operatorname{d\\!}v$
$\displaystyle S_{n,m}$
$\displaystyle\xrightarrow{a.s.}S_{C}=\int_{0}^{1}\int_{0}^{1}\Big{\\{}C(u,v)-C(v,u)\Big{\\}}^{2}\operatorname{d\\!}C(u,v),$
$\displaystyle T_{n,m}$
$\displaystyle\xrightarrow{a.s.}T_{C}=\sup_{(u,v)\in[0,1]^{2}}\left|C(u,v)-C(v,u)\right|.$

Specifically, under the alternative hypothesis, as $n\to\infty$,

$\displaystyle nR_{n,m}\xrightarrow{a.s.}\infty,\quad
nS_{n,m}\xrightarrow{a.s.}\infty,\quad\sqrt{n}T_{n,m}\xrightarrow{a.s.}\infty,$

which ensures the consistency of the proposed test statistics.

###### Proof of Theorem 3.

The proof is postponed to the appendix. ∎

######  Remark 1.

Based on arguments in (Segers et al., 2017, Proposition 3.4), Theorem 1-3 are
valid under Assumption 1-2 with a weaker condition on $m$, that is,
$\,n/m^{2}\rightarrow c\geq 0$.

##  3 Multiplier bootstrap

As shown in preceding section, the asymptotic limits of the statistics are
functions of the unknown $C$ copula, therefore it is impossible to compute
valid P-values using standard Monte Carlo procedure directly. To overcome this
issue, different bootstrap methods were developed, see for example Genest et
al. (2009) and Belalia et al. (2017) among others. However, this approach is
computationally intensive, especially when the sample size is large. The
multiplier bootstrap is an alternative methodology that mitigates the burden
of computation. It estimates the replicates of statistics under null
hypothesis straightly, for the application of this method, the reader is
directed to Rémillard and Scaillet (2009), Genest et al. (2012), Bahraoui et
al. (2018) and reference therein.

Following the multiplier procedure described in Harder and Stadtmüller (2017),
let $H\in\mathbb{N}$ and for each $h\in\\{1,\ldots,H\\}$, let
$\bm{\xi}_{n}^{(h)}=\left(\xi_{1}^{(h)},\ldots,\xi_{n}^{(h)}\right)$ be a
vector of independent random variables with unit mean and unit variance
(taking $\xi_{i}^{(h)}\sim\rm{Exp}(1)$, $i=1,\ldots,n$). Set

$C_{n,m}(u,v)=\frac{1}{n}\sum_{i=1}^{n}\sum_{k=0}^{m}\sum_{\ell=0}^{m}{\mathbb{I}}\left(U_{i}\leq\frac{k}{m},V_{i}\leq\frac{\ell}{m}\right)P_{m,k}(u)P_{m,\ell}(v),$

where
$\left(U_{i},V_{i}\right)=\left(F\left(X_{i}\right),G\left(Y_{i}\right)\right)$.
Denote the sample mean of $\bm{\xi}_{n}^{(h)}$ by $\bar{\xi}_{n}^{(h)}$, then
for each $h$, define

$\displaystyle\widetilde{\mathbb{B}}_{n,m}(u,v)$
$\displaystyle=n^{1/2}\big{\\{}C_{n,m}(u,v)-C(u,v)\big{\\}}$
$\displaystyle=n^{1/2}\Bigg{\\{}\frac{1}{n}\sum_{i=1}^{n}\sum_{k=0}^{m}\sum_{\ell=0}^{m}\bigg{\\{}{\mathbb{I}}\left(U_{i}\leq\frac{k}{m},V_{i}\leq\frac{\ell}{m}\right)-C(u,v)\bigg{\\}}P_{m,k}(u)P_{m,\ell}(v)\Bigg{\\}},$

and

$\displaystyle\overline{\mathbb{B}}_{n,m}^{(h)}(u,v)$
$\displaystyle=n^{1/2}\Bigg{\\{}\frac{1}{n}\sum_{i=1}^{n}\sum_{k=0}^{m}\sum_{\ell=0}^{m}\left(\xi^{(h)}_{i}-\bar{\xi}_{n}^{(h)}\right){\mathbb{I}}\left(\widehat{U}_{i}\leq\frac{k}{m},\widehat{V}_{i}\leq\frac{\ell}{m}\right)P_{m,k}(u)P_{m,\ell}(v)\Bigg{\\}}.$
(6)

One can observe that $\widetilde{\mathbb{B}}_{n,m}$ is the empirical Bernstein
copula process when margins are known. The following proposition states that
$\widetilde{\mathbb{B}}_{n,m}$ converges weakly to the $C$-Brownian bridge
$\mathbb{C}$ mentioned in (5), moreover, $\overline{\mathbb{B}}_{n,m}^{(h)}$
is a valid replicate of $\widetilde{\mathbb{B}}_{n,m}$.

######  Proposition 1.

Let $C$ be a symmetric copula satisfying Assumption 1-2, if
$n/m^{2}\rightarrow c\geq 0$, then for $(u,v)\in[0,1]^{2}$,

  1. 1.

we have

$\widetilde{\mathbb{B}}(u,v)\rightsquigarrow\mathbb{C}(u,v),$

  2. 2.

also, for each $h\in\\{1,\ldots,H\\}$,

$\overline{\mathbb{B}}_{n,m}^{(h)}(u,v)\rightsquigarrow\mathbb{C}(u,v).$ (7)

###### Proof of Proposition 1.

The proof is postponed to the appendix. ∎

Hence the bootstrap replicates of $\mathbb{B}_{n,m}(u,v)$ for
$h\in\\{1,\ldots,H\\}$ are

$\mathbb{B}_{n,m}^{(h)}(u,v)=\overline{\mathbb{B}}^{(h)}_{n,m}(u,v)-\frac{\partial\widehat{C}_{n,m}(u,v)}{\partial
u}\overline{\mathbb{B}}^{(h)}_{n,m}(u,1)-\frac{\partial\widehat{C}_{n,m}(u,v)}{\partial
v}\overline{\mathbb{B}}^{(h)}_{n,m}(1,v).$

The partial derivatives of the empirical Bernstein copula were studied in
Janssen et al. (2016), more specifically,

$\displaystyle\frac{\partial\widehat{C}_{n,m}(u,v)}{\partial u}$
$\displaystyle=m\sum_{k=0}^{m-1}\sum_{\ell=0}^{m}\bigg{\\{}\widehat{C}_{n}\left(\frac{k+1}{m},\frac{\ell}{m}\right)-\widehat{C}_{n}\left(\frac{k}{m},\frac{\ell}{m}\right)\bigg{\\}}P_{m-1,k}(u)P_{m,\ell}(v),$
$\displaystyle\frac{\partial\widehat{C}_{n,m}(u,v)}{\partial v}$
$\displaystyle=m\sum_{k=0}^{m}\sum_{\ell=0}^{m-1}\bigg{\\{}\widehat{C}_{n}\left(\frac{k}{m},\frac{\ell+1}{m}\right)-\widehat{C}_{n}\left(\frac{k}{m},\frac{\ell}{m}\right)\bigg{\\}}P_{m,k}(u)P_{m-1,\ell}(v).$

Unlike the empirical copula, the partial derivatives of the empirical
Bernstein copula can be calculated directly without any further estimations.
Following proposition provides the uniform consistency of partial derivatives.

######  Proposition 2.

Let $C$ be a copula satisfying Assumption 1-2, if $n/m^{2}\to c\geq 0$, then

$\displaystyle\sup_{(u,v)\in(0,1)^{2}}\left|\frac{\partial\widehat{C}_{n,m}(u,v)}{\partial
u}-\dot{C}_{1}(u,v)\right|$ $\displaystyle=O\left(m^{1/2}n^{-1/2}(\log\log
n)^{1/2}\right),$
$\displaystyle\sup_{(u,v)\in(0,1)^{2}}\left|\frac{\partial\widehat{C}_{n,m}(u,v)}{\partial
v}-\dot{C}_{2}(u,v)\right|$ $\displaystyle=O\left(m^{1/2}n^{-1/2}(\log\log
n)^{1/2}\right),$

almost surely as $n\to\infty$.

###### Proof of Proposition 2.

The proof is postponed to the appendix. ∎

Further, due to the fact that under the null hypothesis

$\dot{C}_{1}(u,v)=\dot{C}_{2}(v,u),$

and applying continuous mapping theorem, the bootstrap replicates for
symmetrised empirical Bernstein copula process are given in the following
theorem.

######  Theorem 4.

Let $C$ be a symmetric copula satisfying Assumption 1-2, if $m=cn^{\alpha}$
with $1/2\leq\alpha<1$ and $c>0$ some constant, then for $(u,v)\in(0,1)^{2}$,

$\left(\mathbb{S}_{n,m},\mathbb{S}_{n,m}^{(1)},\ldots,\mathbb{S}_{n,m}^{(H)}\right)\rightsquigarrow\left(\mathbb{S}_{C},\mathbb{S}_{C}^{(1)},\ldots,\mathbb{S}_{C}^{(H)}\right),$

as $n\rightarrow\infty$, providing that for each $h\in\\{1,\ldots,H\\}$,

$\displaystyle\mathbb{S}_{n,m}^{(h)}(u,v)$
$\displaystyle=\mathbb{B}_{n,m}^{(h)}(u,v)-\mathbb{B}_{n,m}^{(h)}(v,u)$
$\displaystyle=\overline{\mathbb{S}}^{(h)}_{n,m}(u,v)+\frac{\partial\widehat{C}_{n,m}(u,v)}{\partial
u}\overline{\mathbb{S}}^{(h)}_{n,m}(u,1)-\frac{\partial\widehat{C}_{n,m}(u,v)}{\partial
v}\overline{\mathbb{S}}^{(h)}_{n,m}(1,v),$

where
$\overline{\mathbb{S}}^{(h)}_{n,m}(u,v)=\overline{\mathbb{B}}^{(h)}_{n,m}(u,v)-\overline{\mathbb{B}}^{(h)}_{n,m}(v,u)$.

Combining Theorems 2 and 4, the asymptotic properties of replicates of
statistics defined by (2.1) are established in the following corollary.

######  Corollary 1.

Let $C$ be a symmetric copula satisfying Assumption 1-2, if $m=cn^{\alpha}$
with $1/2\leq\alpha<1$ and $c>0$ for some constant, then as
$n\rightarrow\infty$, for all $H\in{\mathbb{N}}$, one has

$\displaystyle\left(nR_{n,m},nR_{n,m}^{(1)},\ldots,nR_{n,m}^{(H)}\right)$
$\displaystyle\rightsquigarrow\left(\mathbb{L}_{R},\mathbb{L}_{R}^{(1)},\ldots,\mathbb{L}_{R}^{(H)}\right),$
$\displaystyle\left(nS_{n,m},nS_{n,m}^{(1)},\ldots,nS_{n,m}^{(H)}\right)$
$\displaystyle\rightsquigarrow\left(\mathbb{L}_{S},\mathbb{L}_{S}^{(1)},\ldots,\mathbb{L}_{S}^{(H)}\right),$
$\displaystyle\left(n^{1/2}T_{n,m},n^{1/2}T_{n,m}^{(1)},\ldots,n^{1/2}T_{n,m}^{(H)}\right)$
$\displaystyle\rightsquigarrow\left(\mathbb{L}_{T},\mathbb{L}_{T}^{(1)},\ldots,\mathbb{L}_{T}^{(H)}\right).$

It follows directly from Corollary 1 that P-values of the proposed tests can
be computed as

$\frac{1}{H}\sum_{h=1}^{H}{\mathbb{I}}\left(R_{n,m}^{(h)}\geq
R_{n,m}\right),\quad\frac{1}{H}\sum_{h=1}^{H}{\mathbb{I}}\left(S_{n,m}^{(h)}\geq
S_{n,m}\right),\quad\frac{1}{H}\sum_{h=1}^{H}{\mathbb{I}}\left(T_{n,m}^{(h)}\geq
T_{n,m}\right).$

This approach will be employed to obtain the empirical level and power as
shown in the next section.

##  4 Finite sample performance

The finite sample performance is investigated in this section. All the tests
were conducted with $500$ repetitions under $\alpha=5\%$ nominal level using
$H=200$ multiplier replicates, also since the statistics shown in Theorem 2
involve integration, a discrete approximation is applied with $20\times 20$
grids on $[0,1]^{2}$. To show the improvement of empirical Bernstein copulas
tests, empirical level and power of the proposed tests were compared with
tests in Genest et al. (2012).

In Table 1, the empirical level of the tests are presented. More precisely,
the tests of Genest et al. (2012) based on the empirical copula are denoted by
$\\{R_{n},S_{n},T_{n}\\}$, the proposed tests based on the empirical Bernstein
copula are denoted by $\\{R_{n,m},S_{n,m},T_{n,m}\\}$. The Gaussian, Clayton,
Gumbel and Frank copulas with Kendall’s tau $\tau=0.25$ were used to generate
samples with sizes $n=\\{50,100,200\\}$. Since the Bernstein order $m$ is
dependent on the sample size, $m=\\{9,13,15\\}$ is chosen as one-to-one
correspondence to $n$. Clearly, all the tests are generally good at keeping
their nominal level.

Table 1: Level of the tests under $\alpha=5\%$ nominal level, are estimated from $500$ samples from symmetric copulas using $H=200$ multiplier bootstrap replicates under Bernstein order $m=\\{9,13,15\\}$ associated with sample size $n=\\{50,100,200\\}$. Model | $(n,m)$ | $R_{n}$ | $R_{n,m}$ | $S_{n}$ | $S_{n,m}$ | $T_{n}$ | $T_{n,m}$
---|---|---|---|---|---|---|---
$\tau=0.25$ |  |  |  |  |  |  |
$\rm{Gaussian}$ | $(50,9)$ | $1.2$ | $2.0$ | $3.2$ | $3.4$ | $1.0$ | $2.0$
$(100,13)$ | $2.6$ | $4.6$ | $4.4$ | $5.2$ | $2.8$ | $4.0$
$(200,15)$ | $2.8$ | $3.4$ | $1.8$ | $3.6$ | $7.4$ | $3.8$
$\rm{Clayton}$ | $(50,9)$ | $1.2$ | $3.2$ | $4.2$ | $4.2$ | $1.0$ | $2.2$
$(100,13)$ | $1.0$ | $3.6$ | $3.6$ | $3.8$ | $1.8$ | $2.6$
$(200,15)$ | $4.4$ | $4.6$ | $4.2$ | $5.2$ | $6.2$ | $4.2$
$\rm{Gumbel}$ | $(50,9)$ | $0.8$ | $3.2$ | $3.4$ | $3.8$ | $0.4$ | $1.6$
$(100,13)$ | $1.8$ | $4.0$ | $2.8$ | $3.4$ | $1.8$ | $2.8$
$(200,15)$ | $3.0$ | $4.2$ | $2.6$ | $4.0$ | $6.2$ | $4.0$
$\rm{Frank}$ | $(50,9)$ | $1.4$ | $2.4$ | $3.4$ | $3.4$ | $1.6$ | $2.2$
$(100,13)$ | $2.6$ | $2.6$ | $2.2$ | $2.6$ | $1.6$ | $2.8$
$(200,15)$ | $3.6$ | $3.8$ | $4.0$ | $4.4$ | $6.4$ | $4.0$

In Tables 2 and 3, samples with $n=100$ and $n=200$ from asymmetric
Gaussian(GA), Frank(FR), Gumbel(GU) and Student(ST) copulas were generated
according to the Khoudraji’s device, respectively. The Khoudraji’s device is
defined as

$K_{\delta}(u,v)=u^{\delta}C(u^{1-\delta},v),$

and is implemented in the `copula` R package .

Different values of the shape parameter $\delta=\\{1/4,1/2,3/4\\}$ as well as
various values of Kendall’s tau $\tau=\\{0.5,0.7,0.9\\}$ were considered to
assess their influence on the power. One can observe that for large value of
$\tau$ or $n$, the power of the tests increase, and they reach their maximum
at $\delta=1/2$. Under all circumstances, the proposed tests outperform the
empirical copulas tests or their differences are negligible. Further, the
improvement of $T_{n}$ is the largest among the three statistics, it indicates
that Bernstein smoothing works better for Kolmogorov-Smirnov type statistic
($T_{n}$) than for Crámer von Mises type statistics ($R_{n},S_{n}$) on these
tests. Finally, combining the table of levels and power, it is safe to claim
that the power of the tests are not affected by the empirical levels.

Selecting the Bernstein order for each test is beyond the scope of this paper.
However, one can plot the graph of Bernstein order in practice to choose an
appropriate one as shown in Figure 1. This figure depicts the power of the
proposed tests as function of the Bernstein polynomial order $m$ for
asymmetric Gumbel copula model with $\delta=3/4,\tau=0.7$. From that figure,
$R_{n,m}$ and $S_{n,m}$ experience a similar pattern as they are both Crámer
von Mises type statistic, the power seem like keeping stable as $m$
increasing. And $T_{n,m}$ decreases when $m$ is large. Overall, they
outperform the empirical copulas tests when $m$ is not extremely small. It is
noted that other asymmetric copulas have shown almost the same pattern, and
are not reported here.

Table 2: Power of the tests, are estimated from $500$ samples from asymmetric
copulas using $H=200$ multiplier bootstrap replicates with Bernstein order
$m=13$ associated with sample size $n=100$.

Model | $\delta$ | $\tau$ | $R_{n}$ | $R_{n,m}$ | $S_{n}$ | $S_{n,m}$ | $T_{n}$ | $T_{n,m}$
---|---|---|---|---|---|---|---|---
$K_{\delta}^{GA}$ | $1/4$ | ${0.5}$ | $5.2$ | $11.2$ | $7.0$ | $11.0$ | $5.0$ | $7.4$
$0.7$ | $31.4$ | $48.2$ | $46.6$ | $51.4$ | $17.0$ | $31.6$
$0.9$ | $90.2$ | $92.8$ | $99.8$ | $98.2$ | $57.0$ | $71.6$
$1/2$ | $0.5$ | $15.6$ | $27.8$ | $18.8$ | $29.6$ | $9.0$ | $18.2$
$0.7$ | $70.2$ | $83.6$ | $75.8$ | $84.0$ | $38.4$ | $67.8$
$0.9$ | $99.6$ | $100.0$ | $100.0$ | $100.0$ | $81.0$ | $98.8$
$3/4$ | $0.5$ | $9.4$ | $18.2$ | $13.8$ | $18.2$ | $6.2$ | $12.2$
$0.7$ | $42.8$ | $62.2$ | $51.0$ | $62.0$ | $23.2$ | $47.4$
$0.9$ | $74.6$ | $88.4$ | $85.2$ | $88.2$ | $44.4$ | $74.0$
$K_{\delta}^{FR}$ | $1/4$ | $0.5$ | $6.0$ | $14.6$ | $11.8$ | $14.0$ | $5.0$ | $8.6$
${0.7}$ | $38.8$ | $55.4$ | $54.4$ | $56.6$ | $22.0$ | $36.2$
$0.9$ | $90.0$ | $90.0$ | $100.0$ | $97.2$ | $60.2$ | $72.2$
$1/2$ | $0.5$ | $21.0$ | $31.6$ | $26.6$ | $32.8$ | $16.4$ | $25.8$
$0.7$ | $75.8$ | $87.4$ | $84.8$ | $89.0$ | $46.4$ | $76.4$
$0.9$ | $99.8$ | $99.8$ | $100.0$ | $100.0$ | $82.0$ | $98.4$
$3/4$ | $0.5$ | $12.2$ | $20.0$ | $14.8$ | $20.6$ | $8.8$ | $14.6$
$0.7$ | $38.4$ | $56.8$ | $46.6$ | $54.8$ | $20.2$ | $44.0$
$0.9$ | $71.0$ | $88.4$ | $81.8$ | $88.6$ | $36.6$ | $70.2$
$K_{\delta}^{GU}$ | $1/4$ | ${0.5}$ | $7.4$ | $15.2$ | $9.8$ | $14.8$ | $8.0$ | $10.6$
$0.7$ | $35.2$ | $50.6$ | $48.2$ | $54.2$ | $18.4$ | $31.4$
$0.9$ | $90.6$ | $86.8$ | $99.8$ | $97.0$ | $58.2$ | $70.6$
$1/2$ | $0.5$ | $20.8$ | $32.2$ | $26.6$ | $35.2$ | $10.2$ | $24.0$
$0.7$ | $80.6$ | $87.6$ | $87.4$ | $89.6$ | $47.6$ | $77.2$
$0.9$ | $100.0$ | $100.0$ | $100.0$ | $100.0$ | $84.6$ | $98.2$
$3/4$ | $0.5$ | $21.8$ | $32.6$ | $24.0$ | $32.2$ | $11.4$ | $21.8$
$0.7$ | $56.0$ | $73.4$ | $62.0$ | $73.0$ | $27.6$ | $57.0$
$0.9$ | $77.0$ | $86.8$ | $86.8$ | $88.0$ | $44.0$ | $75.4$
$K_{\delta}^{ST}$ | $1/4$ | $0.5$ | $5.8$ | $14.4$ | $11.0$ | $17.2$ | $7.4$ | $9.2$
$0.7$ | $36.8$ | $48.0$ | $51.4$ | $54.0$ | $18.6$ | $32.6$
$0.9$ | $88.4$ | $88.8$ | $99.4$ | $95.6$ | $55.0$ | $72.6$
$1/2$ | $0.5$ | $16.0$ | $29.8$ | $24.4$ | $30.2$ | $11.0$ | $20.8$
$0.7$ | $71.0$ | $82.6$ | $80.4$ | $84.4$ | $38.6$ | $68.4$
$0.9$ | $99.8$ | $100.0$ | $100.0$ | $100.0$ | $78.2$ | $97.8$
$3/4$ | $0.5$ | $12.4$ | $23.6$ | $15.4$ | $22.4$ | $8.6$ | $15.8$
$0.7$ | $40.8$ | $57.8$ | $54.0$ | $60.8$ | $22.4$ | $43.8$
$0.9$ | $75.4$ | $87.8$ | $84.0$ | $88.6$ | $42.0$ | $73.2$

Table 3: Power of the tests, are estimated from $500$ samples from asymmetric
copulas using $H=200$ multiplier bootstrap replicates under Bernstein order
$m=15$ associated with sample size $n=200$.

Model | $\delta$ | $\tau$ | $R_{n}$ | $R_{n,m}$ | $S_{n}$ | $S_{n,m}$ | $T_{n}$ | $T_{n,m}$
---|---|---|---|---|---|---|---|---
$K_{\delta}^{GA}$ | $1/4$ | ${0.5}$ | $15.0$ | $24.0$ | $14.8$ | $22.8$ | $13.4$ | $14.6$
$0.7$ | $81.8$ | $88.4$ | $82.6$ | $89.4$ | $59.8$ | $70.4$
$0.9$ | $100.0$ | $100.0$ | $100.0$ | $100.0$ | $100.0$ | $100.0$
$1/2$ | $0.5$ | $37.0$ | $45.4$ | $36.0$ | $47.8$ | $27.0$ | $33.0$
$0.7$ | $98.8$ | $98.8$ | $99.0$ | $99.8$ | $88.2$ | $96.6$
$0.9$ | $100.0$ | $100.0$ | $100.0$ | $100.0$ | $100.0$ | $100.0$
$3/4$ | $0.5$ | $32.4$ | $44.2$ | $32.6$ | $43.0$ | $27.4$ | $32.8$
$0.7$ | $87.8$ | $93.2$ | $90.0$ | $94.0$ | $67.6$ | $84.8$
$0.9$ | $99.8$ | $99.8$ | $99.8$ | $99.8$ | $92.2$ | $98.6$
$K_{\delta}^{FR}$ | $1/4$ | $0.5$ | $25.2$ | $34.4$ | $25.4$ | $35.4$ | $24.2$ | $25.4$
${0.7}$ | $93.2$ | $95.0$ | $95.0$ | $95.6$ | $80.2$ | $84.2$
$0.9$ | $100.0$ | $100.0$ | $100.0$ | $100.0$ | $100.0$ | $99.4$
$1/2$ | $0.5$ | $52.2$ | $60.8$ | $52.0$ | $61.4$ | $41.2$ | $48.2$
$0.7$ | $99.6$ | $100.0$ | $99.6$ | $100.0$ | $95.4$ | $99.0$
$0.9$ | $100.0$ | $100.0$ | $100.0$ | $100.0$ | $100.0$ | $100.0$
$3/4$ | $0.5$ | $32.8$ | $41.8$ | $34.4$ | $42.6$ | $26.4$ | $34.2$
$0.7$ | $79.2$ | $88.4$ | $80.8$ | $89.6$ | $62.8$ | $78.2$
$0.9$ | $99.2$ | $99.6$ | $99.6$ | $99.6$ | $91.6$ | $99.0$
$K_{\delta}^{GU}$ | $1/4$ | ${0.5}$ | $21.2$ | $30.6$ | $21.8$ | $29.2$ | $20.2$ | $20.6$
$0.7$ | $89.4$ | $92.6$ | $89.8$ | $92.2$ | $68.4$ | $79.2$
$0.9$ | $100.0$ | $100.0$ | $100.0$ | $100.0$ | $99.6$ | $99.6$
$1/2$ | $0.5$ | $59.6$ | $67.4$ | $57.8$ | $67.6$ | $45.8$ | $55.6$
$0.7$ | $99.6$ | $99.8$ | $99.6$ | $99.8$ | $94.4$ | $98.4$
$0.9$ | $100.0$ | $100.0$ | $100.0$ | $100.0$ | $100.0$ | $100.0$
$3/4$ | $0.5$ | $54.6$ | $66.2$ | $56.0$ | $65.2$ | $41.4$ | $54.4$
$0.7$ | $94.2$ | $96.4$ | $94.8$ | $96.6$ | $77.2$ | $91.2$
$0.9$ | $99.2$ | $99.4$ | $99.2$ | $99.2$ | $92.6$ | $98.8$
$K_{\delta}^{ST}$ | $1/4$ | $0.5$ | $20.8$ | $25.6$ | $21.4$ | $27.6$ | $16.4$ | $20.2$
$0.7$ | $93.2$ | $95.0$ | $95.0$ | $95.6$ | $80.2$ | $84.2$
$0.9$ | $100.0$ | $100.0$ | $100.0$ | $100.0$ | $100.0$ | $99.4$
$1/2$ | $0.5$ | $52.2$ | $60.8$ | $52.0$ | $61.4$ | $41.2$ | $48.2$
$0.7$ | $99.6$ | $100.0$ | $99.6$ | $100.0$ | $95.4$ | $99.0$
$0.9$ | $100.0$ | $100.0$ | $100.0$ | $100.0$ | $100.0$ | $100.0$
$3/4$ | $0.5$ | $32.8$ | $41.8$ | $34.4$ | $42.6$ | $26.4$ | $34.2$
$0.7$ | $79.2$ | $88.4$ | $80.8$ | $89.6$ | $62.8$ | $78.2$
$0.9$ | $99.2$ | $99.6$ | $99.6$ | $99.6$ | $91.6$ | $99.0$

Figure 1: The power ($\%$) of the proposed tests versus the Bernstein
polynomial order $m$. Left: Gumbel copula with sample size $n=100$; Right:
Gumbel copula with sample size $n=200$.

##  5 Ocean data application

A simple illustration was carried out on the ocean data of the National Data
Buoy Center (NDBC), US. The data were collected at south Kodiak (Station
$46066$). There are three variables of interest: WVHT (significant wave
height), APD (average wave period) and WSPD (wind speed) during the winter
season of $2015$ (from November $2014$ to February $2015$). This data was
analyzed in Zhang et al. (2018) with $3910$ observations. Since only
historical data are available on NDBC, we analyzed the data with $2118$
observations here, which is a subset of Zhang et al. (2018).

The unit for WVHT is in meters, whereas the unit for APD and WSPD is in second
and meter per second, respectively. The goal is to detect the asymmetric
structure between the wave height and the other two variables. Scatter plot is
shown in Figure 2, one can observe that APD and WSPD are positive correlated
with WVHT according to Spearman’s rho. Clearly, as presented in Figure 3,
these data are asymmetric.

Figure 2: Scatter plot and Spearman’s rho between WVHT (significant wave
height), APD (average wave period) and WSPD (wind speed).

Figure 3: Rank plots of the pairs. Left: WVHT (significant wave height) versus APD (average wave period); Right: WVHT (significant wave height) versus WSPD (wind speed). Table 4: P-values of empirical tests and Bernstein tests. Test | $R_{n}$ | $R_{n,m}$ | $S_{n}$ | $S_{n,m}$ | $T_{n}$ | $T_{n,m}$
---|---|---|---|---|---|---
WVHT/APD | $0$ | $0$ | $0.0025$ | $0$ | $0$ | $0$
WVHT/WSPD | $0$ | $0$ | $0.0174$ | $0.015$ | $0$ | $0$

To apply the proposed testing procedure, the Bernstein polynomial order was
taken to be $m=50$, and other settings (the nominal level $\alpha$, number of
multiplier bootstrap replicates $H$ and number of grid $N$) are the same as in
simulation study. From Table 4, the Bernstein tests have smaller P-values than
empirical tests. We conclude that they can capture more easily asymmetric
information from the data than their counterparts.

Once the asymmetric dependence structure is confirmed, all the symmetric
copula family (which is a huge proportion) should be ruled out before
modelling. That is, to construct the copula function of $\rm{(WVHT,APD)}$ and
$\rm{(WVHT,WSPD)}$, only asymmetric copulas can be considered. Naturally, one
can use Khoudraji’s device in Section 4 to obtain asymmetric copula models. In
addition, there are various methods available in the literature, for example
the product method in Liebscher (2008) and the linear convex combination
method in Wu (2014).

##  6 Final remarks

New tests based on Bernstein polynomials for symmetry of bivariate copulas
have been proposed and studied. These tests smoothed the empirical tests in
Genest et al. (2012), which provides better performance in simulation study
and real data application. Limiting distribution of the proposed test
statistics were investigated and a Bernstein version of multiplier bootstrap
was constructed and implemented to simulate P-values.

Since the underlying copula is continuous, smooth copula estimator like
empirical Bernstein copula is competitive with empirical copula. From the
bias-variance trade-off point of view, with appropriate smoothness parameter,
the former can outperform the latter by balancing the bias and variance. On
the other hand, the smoothed Bernstein tests still hold the same features as
the non-smoothed empirical tests. For example, the empirical tests tend to
have empirical level which is below the nominal level, see Bahraoui et al.
(2018), Beare and Seo (2020) and Genest et al. (2012). The proposed Bernstein
tests still undergo this pattern, but are less affected.

For future study, it would be possible to apply the Bernstein polynomials for
testing various kinds of symmetry in Nelsen (1993) and the vertex symmetry,
diametrical symmetry developed in Mangold (2017) recently. Theoretically, any
tests based on empirical copulas can be adapted by Bernstein polynomials and
their performances will be improved.

##  7 Acknowledgements

M. Belalia gratefully acknowledge the research support of the Natural Sciences
and Engineering Research Council of Canada(RGPIN/05496-2020 ).

##  Appendix A Proof of Theorem 2

###### Proof.

This proof is an adaption of Genest et al. (2012, Proposition 3). For the
convergence of $nR_{n,m}$ and $n^{1/2}T_{n,m}$, one can directly apply
continuous mapping theorem combined with Theorem 1. For the convergence of
$nS_{n,m}$, the functional delta method was used. To this end, let
$C[0,1]^{2}$ denote the space of continuous functions on $[0,1]^{2}$,
$D[0,1]^{2}$ denote the space of functions with continuity from upper right
quadrant and limits from other quadrants on $[0,1]^{2}$, equipped with sup-
norm. Further, denote $\rm{BV}_{1}[0,1]^{2}$ by the subspace of $D[0,1]^{2}$
where functions with total variation bounded by $1$. By continuous mapping
theorem,

$\left(\mathbb{S}^{2}_{n,m},\mathbb{B}_{n,m}\right)\rightsquigarrow\left(\mathbb{S}_{C}^{2},\mathbb{B}_{C}\right)$

in the space $\ell^{\infty}[0,1]^{2}\times\ell^{\infty}[0,1]^{2}$. Rewrite it
as

$\left(\mathbb{S}^{2}_{n,m},\mathbb{B}_{n,m}\right)=n^{1/2}\\{(A_{n,m},\widehat{C}_{n,m})-(A,C)\\},$

where $A\equiv 0$ and
$A_{n,m}\mathrel{\mathop{\ordinarycolon}}=n^{1/2}(\widehat{C}_{n,m}-\widehat{C}_{n,m}^{\top})^{2}$,
where $\widehat{C}_{n,m}^{\top}(u,v)=\widehat{C}_{n,m}(v,u)$. Then, consider
the map
$\phi\mathrel{\mathop{\ordinarycolon}}\ell^{\infty}[0,1]^{2}\times\rm{BV}_{1}[0,1]^{2}\to{\mathbb{R}}$
defined by

$\phi(a,b)=\int_{(0,1]^{2}}a\operatorname{d\\!}b,$

Clearly,

$nS_{n,m}=\phi\left(\mathbb{S}^{2}_{n,m},\mathbb{B}_{n,m}\right)=n^{1/2}\\{\phi(A_{n,m},\widehat{C}_{n,m})-\phi(A,C)\\}.$

To conclude the proof, by (Carabarin-Aguirre and Ivanoff, 2010, Lemma 4.3),
$\phi$ is Hadamard differentiable tangentially to $C[0,1]^{2}\times
D[0,1]^{2}$ at each $(\alpha,\beta)$ in
$\ell^{\infty}[0,1]^{2}\times\rm{BV}_{1}[0,1]^{2}$ such that
$\int|d\alpha|<\infty$ with derivative

$\phi^{\prime}_{(\alpha,\beta)}(a,b)=\int\alpha\operatorname{d\\!}b+\int
a\operatorname{d\\!}\beta.$

Then by applying the functional delta method,
$nS_{n,m}\rightsquigarrow\phi^{\prime}_{(A,C)}\left(\mathbb{S}_{C}^{2},\mathbb{B}_{C}\right)$,
where

$\phi^{\prime}_{(A,\,C)}\left(\mathbb{S}_{C}^{2},\mathbb{B}_{C}\right)=\int_{(0,1]^{2}}A\operatorname{d\\!}\mathbb{B}_{C}+\int_{(0,1]^{2}}\mathbb{S}_{C}^{2}\operatorname{d\\!}C=\int_{(0,1]^{2}}\mathbb{S}_{C}^{2}\operatorname{d\\!}C.$

This yields to the desired result. ∎

##  Appendix B Proof of Theorem 3

###### Proof.

This proof is an adaption of Genest et al. (2012, Proposition 4). The strongly
uniform consistency of $\widehat{C}_{n,m}$ was provided in (Janssen et al.,
2012, Theorem 1.), then by continuous mapping theorem, it follows immediately
that $R_{n,m}$ and $T_{n,m}$ converge to $R_{C}$ and $T_{C}$ almost surely,
respectively. Further, to prove the convergence of $S_{n,m}$, write

$|S_{n,m}-S_{C}|\leq|\gamma_{n,m}|+|\zeta_{n}|,$

where

$\displaystyle\gamma_{n,m}$
$\displaystyle=\int_{0}^{1}\int_{0}^{1}\left\\{\widehat{C}_{n,m}(u,v)-\widehat{C}_{n,m}(v,u)\right\\}^{2}\operatorname{d\\!}\widehat{C}_{n}(u,v)$
$\displaystyle\quad-\int_{0}^{1}\int_{0}^{1}\left\\{{C}(u,v)-{C}(v,u)\right\\}^{2}\operatorname{d\\!}\widehat{C}_{n}(u,v),$

and

$\displaystyle\zeta_{n}$
$\displaystyle=\int_{0}^{1}\int_{0}^{1}\left\\{{C}(u,v)-{C}(v,u)\right\\}^{2}\operatorname{d\\!}\widehat{C}_{n}(u,v)-\int_{0}^{1}\int_{0}^{1}\left\\{{C}(u,v)-{C}(v,u)\right\\}^{2}\operatorname{d\\!}{C}(u,v).$

Since

$\displaystyle\left|\left\\{\widehat{C}_{n,m}(u,v)-\widehat{C}_{n,m}(v,u)\right\\}^{2}-\left\\{C(u,v)-C(v,u)\right\\}^{2}\right|$
$\displaystyle=\left|\left[\widehat{C}_{n,m}(u,v)+C(u,v)\right]-\left[\widehat{C}_{n,m}(v,u)+C(v,u)\right]\right|$
$\displaystyle\quad\times\left|\left[\widehat{C}_{n,m}(u,v)-C(u,v)\right]-\left[\widehat{C}_{n,m}(v,u)-C(v,u)\right]\right|$
$\displaystyle\leq\left[\left|\widehat{C}_{n,m}(u,v)+C(u,v)\right|+\left|\widehat{C}_{n,m}(v,u)+C(v,u)\right|\right]$
$\displaystyle\quad\times\left[\left|\widehat{C}_{n,m}(u,v)-C(u,v)\right|+\left|\widehat{C}_{n,m}(v,u)-C(v,u)\right|\right]$
$\displaystyle\leq
8\sup_{(u,v)\in[0,1]^{2}}\left|\widehat{C}_{n,m}(u,v)-C(u,v)\right|,$

one has

$|\gamma_{n,m}|\leq
8\sup_{(u,v)\in[0,1]^{2}}\left|\widehat{C}_{n,m}(u,v)-C(u,v)\right|\xrightarrow{a.s.}0.$

For $\zeta_{n}$, by (Genest et al., 1995, Proposition A.1 (i)), one has

$\int_{0}^{1}\int_{0}^{1}\left\\{{C}(u,v)-{C}(v,u)\right\\}^{2}\operatorname{d\\!}\widehat{C}_{n}(u,v)\rightarrow\int_{0}^{1}\int_{0}^{1}\left\\{{C}(u,v)-{C}(v,u)\right\\}^{2}\operatorname{d\\!}{C}(u,v),$

then $\zeta_{n}\rightarrow 0$. Therefore, $S_{n,m}$ converges to $S_{C}$
almost surely. ∎

##  Appendix C Proof of Proposition 1

###### Proof.

By Rémillard and Scaillet (2009), one has

$\displaystyle\mathbb{C}_{n}(u,v)=\sqrt{n}\left\\{C_{n}(u,v)-C(u,v)\right\\}$
$\displaystyle=n^{1/2}\Bigg{\\{}\frac{1}{n}\sum_{i=1}^{n}\left\\{{\mathbb{I}}\left(U_{i}\leq
u,V_{i}\leq v)-C(u,v\right)\right\\}\Bigg{\\}}$
$\displaystyle\quad\rightsquigarrow\mathbb{C}(u,v),$

where $C_{n}(u,v)=\frac{1}{n}\sum_{i=1}^{n}{\mathbb{I}}\left(U_{i}\leq
u,V_{i}\leq v\right)$, and

$\displaystyle\overline{\mathbb{C}}^{(h)}_{n}(u,v)$
$\displaystyle=n^{1/2}\Bigg{\\{}\frac{1}{n}\sum_{i=1}^{n}\left(\xi^{(h)}_{i}-\bar{\xi}_{n}^{(h)}\right){\mathbb{I}}(\widehat{U}_{i}\leq
u,\widehat{V}_{i}\leq v)\Bigg{\\}}\rightsquigarrow\mathbb{C}(u,v).$

To end the proof, one needs to show that the difference between
$\left(\mathbb{C}_{n},\overline{\mathbb{C}}^{(h)}_{n}\right)$ and
$\left(\widetilde{\mathbb{B}}_{n,m},\overline{\mathbb{B}}^{(h)}_{n,m}\right)$
are asymptotically negligible.

It is well-known (for example, see Deheuvels (1979)) that, as $n\to\infty$,

$\|C_{n}(u,v)-C(u,v)\|=O\left(n^{-1/2}(\log\log n)^{1/2}\right)$

almost surely and using the same techniques in Janssen et al. (2012) gives

$\|C_{n,m}(u,v)-C(u,v)\|=O\left(n^{-1/2}(\log\log n)^{1/2}\right)$

almost surely, it immediately follows that

$\|C_{n,m}(u,v)-C_{n}(u,v)\|=O\left(n^{-1/2}(\log\log n)^{1/2}\right)$

almost surely. Therefore, one can conclude that
$\widetilde{\mathbb{B}}_{n,m}(u,v)\rightsquigarrow\mathbb{C}(u,v)$.

Further, by (Janssen et al., 2012, Theorem 1, Lemma 1), if $n/m^{2}\rightarrow
c\geq 0$, then as $n\to\infty$,

$\displaystyle\|\widehat{C}_{n}(u,v)-C(u,v)\|$
$\displaystyle=O\left(n^{-1/2}(\log\log n)^{1/2}\right),$
$\displaystyle\|\widehat{C}_{n,m}(u,v)-C(u,v)\|$
$\displaystyle=O\left(n^{-1/2}(\log\log n)^{1/2}\right)$

almost surely, hence

$\|\widehat{C}_{n,m}(u,v)-\widehat{C}_{n}(u,v)\|=O\left(n^{-1/2}(\log\log
n)^{1/2}\right)$

almost surely, one has that
$\overline{\mathbb{B}}_{n,m}^{(h)}(u,v)\rightsquigarrow\mathbb{C}(u,v)$. ∎

##  Appendix D Proof of Proposition 2

###### Proof.

We only show the uniform consistency for

$\frac{\partial\widehat{C}_{n,m}(u,v)}{\partial
u}=m\sum_{k=0}^{m-1}\sum_{\ell=0}^{m}\bigg{\\{}\widehat{C}_{n}\left(\frac{k+1}{m},\frac{\ell}{m}\right)-\widehat{C}_{n}\left(\frac{k}{m},\frac{\ell}{m}\right)\bigg{\\}}P_{m-1,k}(u)P_{m,\ell}(v).$

The result for the other partial derivative can be obtained similarly. For any
$(u,v)\in(0,1)^{2}$, one has

$\displaystyle\left|\frac{\partial\widehat{C}_{n,m}(u,v)}{\partial
u}-\dot{C}_{1}(u,v)\right|$
$\displaystyle\quad\leq\Bigg{|}m\sum_{k=0}^{m-1}\sum_{\ell=0}^{m}\bigg{\\{}\widehat{C}_{n}\left(\frac{k+1}{m},\frac{\ell}{m}\right)-\widehat{C}_{n}\left(\frac{k}{m},\frac{\ell}{m}\right)-{C}\left(\frac{k+1}{m},\frac{\ell}{m}\right)+{C}\left(\frac{k}{m},\frac{\ell}{m}\right)\bigg{\\}}P_{m-1,k}(u)$
$\displaystyle\qquad\cdot
P_{m,\ell}(v)\Bigg{|}+\Bigg{|}m\sum_{k=0}^{m-1}\sum_{\ell=0}^{m}\bigg{\\{}{C}\left(\frac{k+1}{m},\frac{\ell}{m}\right)-{C}\left(\frac{k}{m},\frac{\ell}{m}\right)\bigg{\\}}P_{m-1,k}(u)P_{m,\ell}(v)-\dot{C}_{1}(u,v)\Bigg{|}$
$\displaystyle\quad=A_{1}+A_{2}.$

Further,

$\displaystyle A_{1}$
$\displaystyle\leq\sum_{k=0}^{m}\sum_{\ell=0}^{m}\bigg{|}\widehat{C}_{n}\left(\frac{k}{m},\frac{\ell}{m}\right)-{C}\left(\frac{k}{m},\frac{\ell}{m}\right)\bigg{|}P_{m,k}^{\prime}(u)P_{m,\ell}(v)$
$\displaystyle\leq\sup_{(u,v)\in[0,1]^{2}}\bigg{|}\widehat{C}_{n}\left(u,v\right)-{C}\left(u,v\right)\bigg{|}\cdot\sum_{k=0}^{m}\left|P_{m,k}^{\prime}(u)\right|$
$\displaystyle=O\left(m^{1/2}n^{-1/2}(\log\log n)^{1/2}\right)$

almost surely as $n\to\infty$ and where

$\sum_{k=0}^{m}\left|P_{m,k}^{\prime}(u)\right|=O(m^{1/2})$

by (Janssen et al., 2014, Lemma 1).

For $A_{2}$, one has

$\displaystyle A_{2}$
$\displaystyle=\left|m\sum_{k=0}^{m-1}\sum_{\ell=0}^{m}\bigg{\\{}\int_{k/m}^{(k+1)/m}\dot{C}_{1}\left(t,\frac{\ell}{m}\right)dt\bigg{\\}}P_{m-1,k}(u)P_{m,\ell}(v)-\dot{C}_{1}(u,v)\right|$
$\displaystyle=\left|\sum_{k=0}^{m-1}\sum_{\ell=0}^{m}\bigg{\\{}\dot{C}_{1}\left(\frac{k}{m-1},\frac{\ell}{m}\right)\bigg{\\}}P_{m-1,k}(u)P_{m,\ell}(v)-\dot{C}_{1}(u,v)\right.$
$\displaystyle\quad+\left.m\sum_{k=0}^{m-1}\sum_{\ell=0}^{m}\bigg{\\{}\int_{k/m}^{(k+1)/m}\left[\dot{C}_{1}\left(t,\frac{\ell}{m}\right)-\dot{C}_{1}\left(\frac{k}{m-1},\frac{\ell}{m}\right)\right]dt\bigg{\\}}P_{m-1,k}(u)P_{m,\ell}(v)\right|.$

Since $\dot{C}_{1}$ is Lipschitz continuous by Assumption 2, let $L$ denote
Lipschitz constant,

$\displaystyle A_{2}$
$\displaystyle\leq\left|\sum_{k=0}^{m-1}\sum_{\ell=0}^{m}\bigg{\\{}\dot{C}_{1}\left(\frac{k}{m-1},\frac{\ell}{m}\right)\bigg{\\}}P_{m-1,k}(u)P_{m,\ell}(v)-\dot{C}_{1}(u,v)\right|$
$\displaystyle\quad+\left|m\sum_{k=0}^{m-1}\sum_{\ell=0}^{m}\bigg{\\{}\int_{k/m}^{(k+1)/m}\left[\dot{C}_{1}\left(t,\frac{\ell}{m}\right)-\dot{C}_{1}\left(\frac{k}{m-1},\frac{\ell}{m}\right)\right]dt\bigg{\\}}P_{m-1,k}(u)P_{m,\ell}(v)\right|$
$\displaystyle\leq\left|\sum_{k=0}^{m-1}\sum_{\ell=0}^{m}L\bigg{\\{}\left|\frac{k}{m-1}-u\right|+\left|\frac{\ell}{m}-v\right|\bigg{\\}}P_{m-1,k}(u)P_{m,\ell}(v)\right|$
$\displaystyle\quad+\left|m\sum_{k=0}^{m-1}\sum_{\ell=0}^{m}\bigg{\\{}\int_{k/m}^{(k+1)/m}L\left|t-\frac{k}{m-1}\right|dt\bigg{\\}}P_{m-1,k}(u)P_{m,\ell}(v)\right|.$

Then let $B_{1}\sim\textsf{Binomial}(m-1,u),B_{2}\sim\textsf{Binomial}(m,v)$
be Binomial random variables, by Cauchy-Schwartz inequality,

$\displaystyle\left|\sum_{k=0}^{m-1}\sum_{\ell=0}^{m}L\left\\{\left|\frac{k}{m-1}-u\right|+\left|\frac{\ell}{m}-v\right|\right\\}P_{m-1,k}(u)P_{m,\ell}(v)\right|$
$\displaystyle\quad+\left|m\sum_{k=0}^{m-1}\sum_{\ell=0}^{m}\left\\{\int_{k/m}^{(k+1)/m}L\left|t-\frac{k}{m-1}\right|dt\right\\}P_{m-1,k}(u)P_{m,\ell}(v)\right|$
$\displaystyle\leq
L\left|\sqrt{\sum_{k=0}^{m-1}\sum_{\ell=0}^{m}\left|\frac{k}{m-1}-u\right|^{2}P_{m-1,k}(u)P_{m,\ell}(v)\cdot\sum_{k=0}^{m-1}\sum_{\ell=0}^{m}P_{m-1,k}(u)P_{m,\ell}(v)}\right.$
$\displaystyle\quad+\left.\sqrt{\sum_{k=0}^{m-1}\sum_{\ell=0}^{m}\left|\frac{\ell}{m}-v\right|^{2}P_{m-1,k}(u)P_{m,\ell}(v)\cdot\sum_{k=0}^{m-1}\sum_{\ell=0}^{m}P_{m-1,k}(u)P_{m,\ell}(v)}\right|$
$\displaystyle\quad+\left|mL\sum_{k=0}^{m-1}\sum_{\ell=0}^{m}\left\\{\frac{1+2k}{2m^{2}}-\frac{k}{m(m-1)}\right\\}P_{m-1,k}(u)P_{m,\ell}(v)\right|$
$\displaystyle=L\left|\sqrt{{\mathbb{E}}\left[\frac{B_{1}}{m-1}-u\right]^{2}}+\sqrt{{\mathbb{E}}\left[\frac{B_{2}}{m}-v\right]^{2}}\right|$
$\displaystyle\quad+\left|mL\sum_{k=0}^{m-1}\sum_{\ell=0}^{m}\left\\{\frac{1+2k}{2m^{2}}-\frac{k}{m(m-1)}\right\\}P_{m-1,k}(u)P_{m,\ell}(v)\right|$
$\displaystyle=L\left\\{\sqrt{\frac{u(1-u)}{m-1}}+\sqrt{\frac{v(1-v)}{m}}\right\\}+L\left|\frac{1}{2m}-\frac{u}{m}\right|$
$\displaystyle=O\left(m^{-1/2}\right).$

Therefore,

$\sup_{(u,v)\in(0,1)^{2}}\left|\frac{\partial\widehat{C}_{n,m}(u,v)}{\partial
u}-\dot{C}_{1}(u,v)\right|=O\left(m^{1/2}n^{-1/2}(\log\log
n)^{1/2}\right)+O\left(m^{-1/2}\right)$

almost surely as $n\to\infty$, which completes the proof. ∎

## References

  * Bahraoui et al. (2018) Bahraoui, T., T. Bouezmarni, and Q. Jean-Francois (2018).  Testing the symmetry of a dependence structure with a characteristic function.  Dependence Modeling 6, 331–355.
  * Bahraoui and Quessy (2022) Bahraoui, T. and J.-F. Quessy (2022).  Tests of multivariate copula exchangeability based on lévy measures.  Scandinavian Journal of Statistics 49(3), 1215–1243.
  * Beare and Seo (2020) Beare, B. K. and J. Seo (2020).  Randomization tests of copula symmetry.  Econometric Theory 00, 1–39.
  * Belalia et al. (2017) Belalia, M., T. Bouezmarni, F. C. Lemyre, and A. Taamouti (2017).  Testing independence based on bernstein empirical copula and copula density.  Journal of Nonparametric Statistics 29(2), 346–380.
  * Carabarin-Aguirre and Ivanoff (2010) Carabarin-Aguirre, A. and B. G. Ivanoff (2010).  Estimation of a distribution udner generalized censoring.  Journal of Multivariate Analysis 101, 1501–1519.
  * Chen and Huang (2007) Chen, S. X. and T.-M. Huang (2007).  Nonparametric estimation of copula functions for dependence modelling.  The Canadian Journal of Statistics 35(2), 265–282.
  * Cousido-Rocha et al. (2019) Cousido-Rocha, M., J. d. Uña-Álvarez, and J. D. Hart (2019).  A two-sample test for the equality of univariate marginal distributions for high-dimensional data.  Journal of Multivariate Analysis 174, 104537.
  * Deheuvels (1979) Deheuvels, P. (1979).  La fonction de dépendance empirique et ses propriétés: Un test non paramétrique d’indépendance.  Académie royale de Belgique, Bulletin de la classe des sciences 65(5), 274–292.
  * Fermanian et al. (2004) Fermanian, J., D. Radulović, and M. Wegkamp (2004).  Weak convergence of empirical copula processes.  Bernoulli 10, 847–860.
  * Fujii (1989) Fujii, Y. (1989).  Test for the equality of marginal distributions on positively dependent bivariate survival data.  Comunications in Statistics- Simulation and Computation 18(2), 633–642.
  * Genest et al. (1995) Genest, C., K. Ghoudi, and R. L.-P. (1995).  A semiparametric estimation procedure of dependnece parameters in multivariate families of ditributions.  Biometrika 82(3), 543–552.
  * Genest et al. (2017) Genest, C., J. Neslehova, and R. Bruno (2017).  Asymptotics behavior of the empirical multilinear copula process under broad conditions.  Journal of Multivariate Analysis 159, 82–110.
  * Genest et al. (2012) Genest, C., J. Neslehova, and Q. Jean-François (2012).  Tests of symmetry for bivariate copulas.  Ann Inst Stat Math 64, 811–834.
  * Genest et al. (2009) Genest, C., B. Rémillard, and D. Beaudoin (2009).  Goodness-of-fit tests for copulas: A review and a power study.  Insurance: Mathematics and Economics 44, 199–213.
  * Gijbels and Mielniczuk (1990) Gijbels, I. and J. Mielniczuk (1990).  Estimating the density of a copula function.  Communications in Statistics - Theory and Methods 2(19), 445–464.
  * Grimaldi and Serinaldi (2006) Grimaldi, S. and F. Serinaldi (2006).  Asymmetric copula in multivariate flood frequency analysis.  Advances in Water Resources 29, 1155–1167.
  * Harder and Stadtmüller (2017) Harder, M. and U. Stadtmüller (2017).  Testing exchangeability of copulas in arbitrary dimension.  Journal of Nonparametric Statistics 29(1), 40–60.
  * Janssen et al. (2012) Janssen, P., J. Swanepoel, and N. Veraverbeke (2012).  Large sample behavior of the bernstein copula estimator.  Journal of Statistical Planing and Inference. 142, 1189–1197.
  * Janssen et al. (2014) Janssen, P., J. Swanepoel, and N. Veraverbeke (2014).  A note on the asymptotic bevahior of the bernstein estimator of the copula density.  Journal of Multivariate Analysis 124, 480–487.
  * Janssen et al. (2016) Janssen, P., J. Swanepoel, and N. Veraverbeke (2016).  Bernstein estimation for a copula derivative with application to conditional distribution and regression functionals.  Test 25(2), 351–374.
  * Jaser and Min (2020) Jaser, M. and A. Min (2020).  On tests for symmetry and radial symmetry of bivariate copulas towards testing for ellipticity.  Computational Statistics 36, 1–26.
  * Kiriliouk et al. (2021) Kiriliouk, A., J. Segers, and H. Tsukahara (2021).  Resampling procedures with empirical beta copulas.  Pioneering works on extreme value theory, 27–53.
  * Kojadinovic (2022) Kojadinovic, I. (2022).  On stute’s representation for a class of smooth, possibly data-adaptive empirical copula processes.  arXiv preprint, arXiv: 2204.11240.
  * Kojadinovic and Yi (2022) Kojadinovic, I. and B. Yi (2022).  A class of smooth, possibly data-adaptive nonparametric copula estimator containing the empirical beta copula.  arXiv preprint, arXiv: 2106.10726.
  * Liebscher (2008) Liebscher, E. (2008).  Construction of asymmetric multivariate copulas.  Journal of Multivariate Analysis 99(10), 2234–2250.
  * Mangold (2017) Mangold, B. (2017).  New concepts of symmetry for copulas.  FAU Discussion Papers in Economics, 06/2017.
  * Morettin et al. (2010) Morettin, P. A., C. M. C. Toloi, C. Chiann, and J. C. S. de Mirand (2010).  Wavelet-smoothed empirical copula estimators.  Revista Brasileira de Finanças 8(3), 263–281.
  * Nelsen (1993) Nelsen, R. B. (1993).  Some concepts of bivariate symmetry.  Journal of Nonparametric Statistics 3, 95–101.
  * Nelsen (2006) Nelsen, R. B. (2006).  An Introduction to Copulas.  Second edition. Springer, New York, NY.
  * Nelsen (2007) Nelsen, R. B. (2007).  Extremes of nonexchangeability.  Statistical Papers 48, 329–336.
  * Omelka et al. (2009) Omelka, M., I. Gijbels, and N. Veraverbeke (2009).  Improved kernel estimation of copulas: weak convergence and goodness-of-fit testing.  The Annals of Statistics 37(5B), 3023–3058.
  * Rémillard and Scaillet (2009) Rémillard, B. and O. Scaillet (2009).  Testing for equality between two copulas.  Journal of multivariate analysis 100, 377–386.
  * Rüschendorf (1976) Rüschendorf, L. (1976).  Asymptotic distributions of multivariate rank order statistics.  The Annals of Statistics 4, 912–923.
  * Sancetta and Satchell (2004) Sancetta, A. and S. Satchell (2004).  The bernstein copula and its applications to modelling and approximations of multivariate distributions.  Econometric Theory 20(3), 535–562.
  * Segers (2012) Segers, J. (2012).  Asymptotics of empirical copula processes under non-restrictive smoothness assumptions.  Bernoulli 18(3), 764–782.
  * Segers et al. (2017) Segers, J., M. Sibuya, and H. Tsukahara (2017).  The empirical beta copula.  Journal of Multivariate Analysis 155, 35–51.
  * Sklar (1959) Sklar, A. (1959).  Fonctions de répartition à n dimensions e leurs marges.  Publications de l’Institut de Statistique de l’Univiversité de Paris 8, 229–231.
  * Wu (2014) Wu, S. (2014).  Construction of asymmetric copulas and its application in two-dimensional reliability modelling.  European Journal of Operational Research 238, 476–485.
  * Zhang et al. (2018) Zhang, Y., C.-W. Kim, M. Beer, D. Huliang, and C. G. Soares (2018).  Modeling multivariate ocean data using asymmetric copulas.  Coastal Engineering 135, 91–111.