[table]capposition=top

Visualization and Assessment of Copula Symmetry

Cristian F. Jiménez-Varón111Statistics Program, King Abdullah University of
Science and Technology, Thuwal 23955-6900, Saudi Arabia.
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>, Hao Lee222Statistics & Data Science, Dietrich College
of Humanities and Social Sciences, Carnegie Mellon University, United States.,
Marc G. Genton1, and Ying Sun1

August 28, 2024

Abstract

Visualization and assessment of copula structures are crucial for accurately
understanding and modeling the dependencies in multivariate data analysis. In
this paper, we introduce an innovative method that employs functional boxplots
and rank-based testing procedures to evaluate copula symmetry. This approach
is specifically designed to assess key characteristics such as reflection
symmetry, radial symmetry, and joint symmetry. We first construct test
functions for each specific property and then investigate the asymptotic
properties of their empirical estimators. We demonstrate that the functional
boxplot of these sample test functions serves as an informative visualization
tool of a given copula structure, effectively measuring the departure from
zero of the test function. Furthermore, we introduce a nonparametric testing
procedure to assess the significance of deviations from symmetry, ensuring the
accuracy and reliability of our visualization method. Through extensive
simulation studies involving various copula models, we demonstrate the
effectiveness of our testing approach. Finally, we apply our visualization and
testing techniques to two real-world datasets: a nutritional habits survey
with five variables and wind speed data from three locations in Saudi Arabia.

Key words: Copula structure; Functional boxplot; Rank-based testing; Symmetry;
Visualization.

##  1 Introduction

Copula models have gained significant prominence in the field of statistics
and data analysis due to their flexibility in modeling intricate dependence
structures among random variables (Nelsen, 2006; Joe, 2014; Patton, 2012).
They have become indispensable tools for capturing and understanding various
types of dependencies, such as tail dependence, asymmetry, and nonlinearity
(Genest and Favre, 2007; Cherubini et al., 2004). By decoupling the marginal
distributions from the dependence structure, copula models offer a powerful
framework for accurately characterizing complex multivariate relationships
(Cherubini et al., 2004; Joe, 2014). In addition to their theoretical
significance, copula models have found extensive real-world applications in
finance, insurance, and environmental sciences. In finance, for instance,
copula models facilitate portfolio optimization, risk management, and pricing
of complex financial derivatives by accurately modeling dependencies between
financial assets (Cherubini et al., 2004; Patton, 2012).

According to the representation theorem provided by Sklar (1959), every
multivariate cumulative distribution function, $F$, of a continuous random
vector $\mathbf{X}=(X_{1},\ldots,X_{d})^{\top}$ on $\mathbb{R}^{d}$, can be
written as

$F(x_{1},\ldots,x_{d})=\mathds{P}(X_{1}\leq x_{1},\ldots,X_{d}\leq
x_{d})=C\\{F_{1}(x_{1}),\ldots,F_{d}(x_{d})\\},$ (1)

where $F_{l}(x_{l})=\mathds{P}(X_{l}\leq x_{l})$, $x_{l}\in\mathbb{R}$, are
the continuous marginal distributions, $l=1,\ldots,d$, and
$C:[0,1]^{d}\rightarrow[0,1]$ is the unique copula that characterizes the
dependence structure of the random vector $\mathbf{X}$ and can be obtained
from

$C(u_{1},\ldots,u_{d})=\mathds{P}(U_{1}\leq u_{1},\ldots,U_{d}\leq
u_{d})=F\\{F_{1}^{-1}(u_{1}),\ldots,F_{d}^{-1}(u_{d})\\},$ (2)

where $F_{l}^{-1}(u_{l})=\text{inf}\\{x|F_{l}(x)\geq u_{l}\\}$,
$u_{l}\in[0,1]$, is the quantile function of $F_{l}$ and
$\mathbf{U}=(U_{1},\ldots,U_{d})^{\top}$ with $U_{l}=F_{l}(X_{l})$. From (2),
the copula function $C$ serves as a cumulative distribution function for the
random vector $\mathbf{U}$, residing within the $d$-dimensional unit hypercube
and characterized by its marginal distributions. In practical applications,
the representation provided in Equation (1) enables the modeling of the
dependence structure, given the knowledge of the marginal distributions. This
can be achieved by selecting an appropriate parametric copula model from a
wide range of options available in the literature (see e.g., Nelsen, 2006;
Joe, 2014).

Choosing an appropriate copula model is a challenging task when quantifying
dependence. In various practical applications, such as actuarial science,
finance, and survival analysis (Nelsen, 2006; Patton, 2006; Aas et al., 2009),
the common approach has been to rely on expert knowledge or choose a copula
model based on mathematical convenience rather than its suitability for the
specific data application. However, this approach can introduce limitations
and biases in the analysis (Mikosch, 2006; Nelsen, 2006; Aas et al., 2009;
Joe, 2014).

Several existing approaches in the literature for copula model selection are
based on goodness-of-fit tests for copulas (Genest and Favre, 2007; Genest et
al., 2009; Berg, 2009). These methods typically treat the univariate marginal
distribution as an infinite-dimensional nuisance parameter and replace the
observations with maximally invariant statistics, such as ranks.

Understanding the properties and structure of copulas is crucial for capturing
and interpreting the relationships between random variables. Various methods
have been proposed in the literature to specify and test copula structures.
Jaworski (2010) introduced a test for the associativity structure of copulas
based on the asymptotic distribution of the pointwise copula estimator.
However, this test only assesses associativity at a specific point rather than
for the entire copula process, as discussed in Bücher et al. (2012). Bücher et
al. (2012) derived Cramér-von Mises and Kolmogorov-Smirnov type test
statistics for evaluating the characteristics of associativity. Additionally,
they developed test statistics for Archimedean copulas (Bücher et al., 2012).
Bücher et al. (2011) proposed a test for extreme value dependence based on the
minimum weighted $L^{2}$-distance of extreme-value copulas. The bivariate
symmetry test for copulas, based on Cramér-von Mises and Kolmogorov-Smirnov
functionals of the rank-based empirical copula process, was introduced by
Genest et al. (2012) and Genest and Nešlehová (2014).

Li and Genton (2013) proposed a nonparametric method for identifying copula
symmetry using the asymptotic distribution of the empirical copula process.
Quessy (2016) developed a statistical framework based on quadratic functionals
to test the identity of copulas from a multivariate distribution. More
recently, Jaser and Min (2021) proposed simpler nonparametric tests for the
symmetry and radial symmetry of bivariate copulas. Their approach involves
creating two bivariate samples by manipulating the underlying copula while
preserving its dependence structure. The test statistics are based on the
difference between the empirical Kendall’s tau of both samples.

In this paper, we present a new approach for visualizing and testing the
structure of copula models, specifically focusing on properties such as
symmetry, radial symmetry, and joint symmetry as defined in Nelsen (1993). Our
approach complements existing goodness-of-fit tests for copula model
selection. To visualize these copula structures, we employ the functional
boxplot introduced by Sun and Genton (2011) as a visual tool to quantify the
deviations from a given copula structure by measuring the departure from zero
of sample test functions. We demonstrate that these visualizations offer
insights into the extent to which specific copula structures are adhered to.

Additionally, we introduce a nonparametric testing procedure to assess the
significance of deviations from symmetry. This testing procedure is motivated
by the techniques proposed by Huang and Sun (2019) and Huang et al. (2023),
which utilize a functional data framework to visualize and assess spatio-
temporal covariance properties in both univariate and multivariate cases. We
evaluate the effectiveness of our proposed testing approach through extensive
simulation studies involving various copula models.

The paper is structured as follows. In Section 2, we outline the copula
symmetry of interest, including reflection symmetry, radial symmetry, and
joint symmetry. We also provide details on the visualization and nonparametric
testing procedures for each of these copula structures. Section 3 presents the
simulation results regarding the size and power of our proposed nonparametric
test. In Section 4, we apply our methods to two real-world datasets: a
nutritional habits survey with five variables and wind speed data from three
locations in Saudi Arabia. Finally, the paper concludes with a discussion in
Section 5.

##  2 Methodology

In Section 2.1, we introduce copula symmetries. Section 2.2 covers the
construction of test functions and provides asymptotic results for proper
estimators. We present the visualization of test functions with functional
boxplots in Section 2.3. Lastly, in Section 2.4, we describe a rank-based
testing procedure for copula symmetry.

###  2.1 Copula Symmetry

Our discussion centers on the symmetry of bivariate copulas. Unlike the case
of univariate functions, the concept of symmetry is not uniquely defined in a
multivariate setting. Therefore, different notions of symmetry have been
investigated in the context of copulas. Here we focus on the ones presented in
Nelsen (1993).

######  Definition 1.

A copula $C$ is said to be symmetric if

$C(u,v)-C(v,u)=0,\hskip 14.22636pt\forall(u,v)\in[0,1]^{2}.$ (3)

Based on the algebraic Equation (3), one should notice that for any symmetric
copula $C$, its distribution is symmetric with respect to the diagonal
connecting the origin and the point $(1,1)$. Thus, the symmetry in Definition
1 is called reflection symmetry in some literature. We will also use
reflection symmetry to refer to this type of symmetry in the sequel.

######  Definition 2.

A copula $C$ is said to be radially symmetric if

$C(u,v)-C(1-u,1-v)+1-u-v=0,\hskip 14.22636pt\forall(u,v)\in[0,1]^{2}.$ (4)

Equivalently, one can state that the radial symmetry property as
$C(u,v)-C^{\ast}(u,v)=0$ for all $(u,v)\in[0,1]^{2}$, where $C^{\ast}$ stands
for the survival copula associated with $C$, i.e., for all
$(u,v)\in[0,1]^{2}$, $C^{\ast}(u,v)=C(1-u,1-v)-1+u+v$. Pointed out by Nelsen
(1993), there exist copulas that are reflection symmetric but not radially
symmetric and, conversely, copulas that are radially symmetric but not
reflection symmetric.

######  Definition 3.

A copula $C$ is said to be jointly symmetric if it satisfies

$C(u,v)+C(u,1-v)-u=0$

and

$C(u,v)+C(1-u,v)-v=0$

for all $(u,v)\in[0,1]^{2}$.

One can easily show that joint symmetry implies radial symmetry. However,
there is no implication between reflection symmetry and joint symmetry. The
Figure 1 of Li and Genton (2013) provides the interrelations among the three
types of symmetry.

###  2.2 Test functions

We propose to assess and visualize the three types of copula symmetries by the
construction of test functions. First, we focus on the construction of test
functions specifically for reflection symmetry (S). For any fixed $v\in[0,1]$,
define the reflection symmetry test functions
$f^{{}^{S}}_{v},g^{{}^{S}}_{v}:[0,1]\rightarrow[-1,1]$ by

$f^{{}^{S}}_{v}(t)=C(t,v)-C(v,t)\text{\hskip 14.22636ptand\hskip
14.22636pt}g^{{}^{S}}_{v}(t)=-f^{{}^{S}}_{v}(t)$

for all $t\in[0,1]$. If $C$ is reflection symmetric, we have
$f^{{}^{S}}_{v}=g^{{}^{S}}_{v}=0$; otherwise, the values of $f^{{}^{S}}_{v}$
and $g^{{}^{S}}_{v}$ vary with respect to $t$. For all $m\in\mathbb{N}$, any
set $\\{v_{1},\ldots,v_{m},w_{1},\ldots,w_{m}\\}\subseteq[0,1]$ can, in the
same manner, induce $2m$ reflection symmetry test functions
$f^{{}^{S}}_{v_{1}},\ldots,f^{{}^{S}}_{v_{m}}$ and
$g^{{}^{S}}_{w_{1}},\ldots,g^{{}^{S}}_{w_{m}}$.

In practice, we need proper estimators $\widehat{f}^{{}^{\hskip
1.13809ptS}}_{v_{1}},\ldots,\widehat{f}^{{}^{\hskip 1.13809ptS}}_{v_{m}}$ and
$\widehat{g}^{{}^{\hskip 1.42262ptS}}_{w_{1}},\ldots,\widehat{g}^{{}^{\hskip
1.13809ptS}}_{w_{m}}$ of the test functions
$f^{{}^{S}}_{v_{1}},\ldots,f^{{}^{S}}_{v_{m}}$ and
$g^{{}^{S}}_{w_{1}},\ldots,g^{{}^{S}}_{w_{m}}$. Intuitively, the estimators
can be obtained through a linear combination of the corresponding empirical
copulas. To provide a clearer definition of these estimators, we briefly
summarize the important asymptotic results of the two-dimensional empirical
copula process (see, e.g., Deheuvels, 1979; Stute, 1984; Tsukahara, 2005).
Additional recent results on convergence rate can be found in Genest and
Segers (2010), Segers (2012), Swanepoel and Allison (2013), and the references
therein.

Let $C$ be a $2$-dimensional copula and $\bm{U}=(U_{1},U_{2})^{\top}$ be a
random vector that is $C$-distributed. If one has direct access to a random
sample $\bm{U}_{1},\ldots,\bm{U}_{n}$ of size $n$, the copula $C$ can be
estimated by the empirical copula process defined by

$\widehat{C}_{n}(u,v)=\frac{1}{n}\sum_{i=1}^{n}\mathbbm{1}\big{\\{}U_{i1}\leq
u,U_{i2}\leq v\big{\\}},\hskip 14.22636pt\forall(u,v)\in[0,1]^{2},$

where $\mathbbm{1}(\cdot)$ denotes the indicator function. As a well-known
result, we have that for all $(u,v)\in[0,1]^{2}$,

$\mathbb{C}_{n}(u,v):=\sqrt{n}\left\\{\widehat{C}_{n}(u,v)-C(u,v)\right\\}\xrightarrow[\hskip
14.22636pt]{d}\mathbb{C}(u,v)\text{\hskip 5.69054ptin\hskip
5.69054pt}\ell^{\infty}([0,1]^{2}),\hskip 14.22636ptn\rightarrow\infty,$ (5)

where $\ell^{\infty}([0,1]^{2})$ denotes the space of all the bounded
functions over the compact set $[0,1]^{2}$ and $\mathbb{C}$ is a
$2$-dimensional pinned $C$-Brownian sheet, i.e., it is a centered Gaussian
random field with the covariance function given by

$\mathrm{Cov}\\{\mathbb{C}(u,v),\hskip 2.84526pt\mathbb{C}(x,y)\\}=C(u\land
x,v\land y)-C(u,v)C(x,y),\hskip 14.22636pt\forall u,v,x,y\in[0,1]$

where for all $a,b\in\mathbb{R}$, $a\land b=\min\\{a,b\\}$ (Genest et al.,
2012). However, it is often the case that the $n$ observations we have are
generated from a random vector $\bm{X}=(X_{1},X_{2})^{\top}$ that is not
necessarily uniformly distributed over the interval $[0,1]$. The
representation theorem (Sklar, 1959) states that it can be expressed as the
composition of a copula $C$ and marginals of $\bm{X}$. In this case, from
every $\bm{X}_{i}$ in the random sample, one can estimate a $\bm{U}_{i}$ by
the pseudo-observation
$\widehat{\bm{U}}_{i}=(\widehat{U}_{i1},\widehat{U}_{i2})^{\top}$, where for
all $s\in\\{1,2\\}$,

$\widehat{U}_{is}=\frac{1}{n}\sum_{r=1}^{n}\mathbbm{1}\big{\\{}X_{rs}\leq
X_{is}\big{\\}}.$

With all of these $\widehat{\bm{U}}_{i}$’s, one can estimate $C$ by

$\widehat{D}_{n}(u,v)=\frac{1}{n}\sum_{i=1}^{n}\mathbbm{1}\big{\\{}\widehat{U}_{i1}\leq
u,\widehat{U}_{i2}\leq v\big{\\}},\hskip 14.22636pt\forall(u,v)\in[0,1]^{2}.$

It has been shown that when $C$ is regular (see Definition 1 in Genest et al.
(2012) for example), or loosely speaking, when $C$ is differentiable with
continuous partials, we have

$\widehat{\mathbb{D}}_{n}(u,v):=\sqrt{n}\Big{\\{}\widehat{D}_{n}(u,v)-C(u,v)\Big{\\}}\\\
\xrightarrow[\hskip
14.22636pt]{d}\mathbb{C}(u,v)-\dot{C}_{1}(u,v)\mathbb{C}(u,1)-\dot{C}_{2}(u,v)\mathbb{C}(1,v)\text{\hskip
5.69054ptin\hskip 5.69054pt}\ell^{\infty}([0,1]^{2}),\hskip
14.22636ptn\rightarrow\infty,$ (6)

where $\dot{C}_{1}$ and $\dot{C}_{2}$ denote the partial derivatives of the
copula $C$ with respect to its first and second variables, respectively. In
the sequel, if not otherwise stated, we always impose the regularity
assumption on the underlying copula $C$. To sum up, the estimator of the
reflection symmetry test function $\widehat{f}^{{}^{\hskip 1.42262ptS}}_{v}$
can be defined by either

$\widehat{f}^{{}^{\hskip
1.42262ptS}}_{v}(t)=\widehat{C}_{n}(t,v)-\widehat{C}_{n}(v,t),\hskip
14.22636pt\forall t\in[0,1],$

or

$\widehat{f}^{{}^{\hskip
1.42262ptS}}_{v}(t)=\widehat{D}_{n}(t,v)-\widehat{D}_{n}(v,t),\hskip
14.22636pt\forall t\in[0,1],$

depending on the type of the data set we have, and we simply set
$\widehat{g}^{{}^{\hskip 1.13809ptS}}_{v}(t)=-\widehat{f}^{{}^{\hskip
1.42262ptS}}_{v}(t)$.

######  Proposition 1.

Given any fixed $v\in[0,1]$, the estimator $\widehat{f}^{{}^{\hskip
1.42262ptS}}_{v}(t)$ satisfies that for all $t\in[0,1]$,

$\sqrt{n}\left\\{\widehat{f}^{{}^{\hskip
1.42262ptS}}_{v}(t)-f^{{}^{S}}_{v}(t)\right\\}\xrightarrow[\hskip
14.22636pt]{d}\begin{dcases}\mathbb{E}^{{}^{S}}_{v}(t)&\text{if
}\widehat{f}^{{}^{\hskip
1.42262ptS}}_{v}(t)=\widehat{C}_{n}(t,v)-\widehat{C}_{n}(v,t)\\\
\widehat{\mathbb{E}}^{{}^{S}}_{v}(t)&\text{if }\widehat{f}^{{}^{\hskip
1.42262ptS}}_{v}(t)=\widehat{D}_{n}(t,v)-\widehat{D}_{n}(v,t)\\\ \end{dcases}$

in $\ell^{\infty}([0,1])$ as $n\rightarrow\infty$, where
$\mathbb{E}^{{}^{S}}_{v}$ and $\widehat{\mathbb{E}}^{{}^{S}}_{v}$ are two
centered Gaussian random fields.

The proof of this Proposition can be found in Appendix A.1.

Type | Access | Test Functions
---|---|---
S | $(U_{1},U_{2})$ | $\widehat{f}^{{}^{\hskip 1.42262ptS}}_{v}(t)=\widehat{C}_{n}(t,v)-\widehat{C}_{n}(v,t),\hskip 14.22636pt\widehat{g}^{{}^{\hskip 1.13809ptS}}_{v}(t)=-\widehat{f}^{{}^{\hskip 1.42262ptS}}_{v}(t)$
$(X_{1},X_{2})$ | $\widehat{f}^{{}^{\hskip 1.42262ptS}}_{v}(t)=\widehat{D}_{n}(t,v)-\widehat{D}_{n}(v,t)\hskip 14.22636pt\widehat{g}^{{}^{\hskip 1.13809ptS}}_{v}(t)=-\widehat{f}^{{}^{\hskip 1.42262ptS}}_{v}(t)$
R | $(U_{1},U_{2})$ | $\widehat{f}^{{}^{\hskip 1.42262ptR}}_{v}(t)=\widehat{C}_{n}(t,v)-\widehat{C}_{n}(1-t,1-v)+1-t-v,\hskip 14.22636pt\widehat{g}^{{}^{\hskip 1.13809ptR}}_{v}(t)=-\widehat{f}^{{}^{\hskip 1.42262ptR}}_{v}(t)$
$(X_{1},X_{2})$ | $\widehat{f}^{{}^{\hskip 1.42262ptR}}_{v}(t)=\widehat{D}_{n}(t,v)-\widehat{D}_{n}(1-t,1-v)+1-t-v,\hskip 14.22636pt\widehat{g}^{{}^{\hskip 1.13809ptR}}_{v}(t)=-\widehat{f}^{{}^{\hskip 1.42262ptR}}_{v}(t)$
J | $(U_{1},U_{2})$ | $\widehat{f}^{{}^{\hskip 1.42262ptJ,1}}_{v}(t)=\widehat{C}_{n}(t,v)+\widehat{C}_{n}(t,1-v)-t,\hskip 14.22636pt\widehat{g}^{{}^{\hskip 1.13809ptJ,1}}_{v}(t)=-\widehat{f}^{{}^{\hskip 1.42262ptJ,1}}_{v}(t)$
$\widehat{f}^{{}^{\hskip
1.42262ptJ,2}}_{v}(t)=\widehat{C}_{n}(t,v)+\widehat{C}_{n}(1-t,v)-v,\hskip
14.22636pt\widehat{g}^{{}^{\hskip
1.13809ptJ,2}}_{v}(t)=-\widehat{f}^{{}^{\hskip 1.42262ptJ,2}}_{v}(t)$
$(X_{1},X_{2})$ | $\widehat{f}^{{}^{\hskip 1.42262ptJ,1}}_{v}(t)=\widehat{D}_{n}(t,v)+\widehat{D}_{n}(t,1-v)-t,\hskip 14.22636pt\widehat{g}^{{}^{\hskip 1.13809ptJ,1}}_{v}(t)=-\widehat{f}^{{}^{\hskip 1.42262ptJ,1}}_{v}(t)$
$\widehat{f}^{{}^{\hskip
1.42262ptJ,2}}_{v}(t)=\widehat{D}_{n}(t,v)+\widehat{D}_{n}(t,1-v)-t,\hskip
14.22636pt\widehat{g}^{{}^{\hskip
1.13809ptJ,2}}_{v}(t)=-\widehat{f}^{{}^{\hskip 1.42262ptJ,2}}_{v}(t)$
Table 1: Definitions of the estimators of the test functions for different
types of symmetry.

Similarly, we can construct radial symmetry (R) test functions, denoted as
$\widehat{f}^{{}^{\hskip 1.42262ptR}}_{v}(t)$ and $\widehat{g}^{{}^{\hskip
1.13809ptR}}_{v}(t)$, based on Definition 2 to estimate $f^{{}^{R}}_{v}(t)$
and $g^{{}^{R}}_{v}(t)$, as described in Table 1. The convergence of these
estimators is presented in Proposition 2.

######  Proposition 2.

Given any fixed $v\in[0,1]$, the estimator $\widehat{f}^{{}^{\hskip
1.42262ptR}}_{v}(t)$ satisfies that for all $t\in[0,1]$,

$\sqrt{n}\left\\{\widehat{f}^{{}^{\hskip
1.42262ptR}}_{v}(t)-f^{{}^{R}}_{v}(t)\right\\}\xrightarrow[\hskip
14.22636pt]{d}\begin{dcases}\mathbb{E}^{{}^{R}}_{v}(t)&\text{if
}\widehat{f}^{{}^{\hskip
1.42262ptR}}_{v}(t)=\widehat{C}_{n}(t,v)-\widehat{C}_{n}(1-t,1-v)+1-t-v\\\
\widehat{\mathbb{E}}^{{}^{R}}_{v}(t)&\text{if }\widehat{f}^{{}^{\hskip
1.42262ptR}}_{v}(t)=\widehat{D}_{n}(t,v)-\widehat{D}_{n}(1-t,1-v)+1-t-v\end{dcases}$

in $\ell^{\infty}([0,1])$ as $n\rightarrow\infty$, where
$\mathbb{E}^{{}^{R}}_{v}$ and $\widehat{\mathbb{E}}^{{}^{R}}_{v}$ are two
centered Gaussian random fields.

The proof of this Proposition can be found in Appendix A.2.

As for the test functions for joint symmetry (J), one should investigate the
two properties in Definition 3 separately. Hence, for any given $v\in[0,1]$,
we construct four population joint symmetry test functions:
$f^{{}^{J,1}}_{v},f^{{}^{J,2}}_{v},g^{{}^{J,1}}_{v},g^{{}^{J,2}}_{v}:[0,1]\rightarrow[-1,1]$
that are given by

$f^{{}^{J,1}}_{v}(t)=C(t,v)+C(t,1-v)-t\text{\hskip 14.22636pt and \hskip
14.22636pt}g^{{}^{J,1}}_{v}(t)=-f^{{}^{J,1}}_{v}(t)$

as well as

$f^{{}^{J,2}}_{v}(t)=C(1-t,v)+C(t,1-v)-v\text{\hskip 14.22636pt and \hskip
14.22636pt}g^{{}^{J,2}}_{v}(t)=-f^{{}^{J,2}}_{v}(t)$

for all $t\in[0,1]$. As the cases of other symmetries, they can be estimated
by the test functions involving empirical copulas (see Table 1 for their
definitions), whose relevant asymptotic results are presented in Proposition
3.

######  Proposition 3.

Given $v\in[0,1]$, the estimator $\widehat{f}^{{}^{\hskip
1.42262ptJ,1}}_{v}(t)$ satisfies that for all $t\in[0,1]$,

$\sqrt{n}\left\\{\widehat{f}^{{}^{\hskip
1.42262ptJ,1}}_{v}(t)-f^{{}^{J,1}}_{v}(t)\right\\}\xrightarrow[\hskip
14.22636pt]{d}\begin{dcases}\mathbb{E}^{{}^{J,1}}_{v}(t)&\text{if
}\widehat{f}^{{}^{\hskip
1.42262ptJ,1}}_{v}(t)=\widehat{C}_{n}(t,v)+\widehat{C}_{n}(t,1-v)-t\\\
\widehat{\mathbb{E}}^{{}^{J,1}}_{v}(t)&\text{if }\widehat{f}^{{}^{\hskip
1.42262ptJ,1}}_{v}(t)=\widehat{D}_{n}(t,v)+\widehat{D}_{n}(t,1-v)-t\end{dcases}$

in $\ell^{\infty}([0,1])$ as $n\rightarrow\infty$, where
$\mathbb{E}^{{}^{J,1}}_{v}$ and $\widehat{\mathbb{E}}^{{}^{J,1}}_{v}$ are two
centered Gaussian random fields.

Similarly, given $v\in[0,1]$, the estimator $\widehat{f}^{{}^{\hskip
1.42262ptJ,2}}_{v}(t)$ satisfies that for all $t\in[0,1]$,

$\sqrt{n}\left\\{\widehat{f}^{{}^{\hskip
1.42262ptJ,2}}_{v}(t)-f^{{}^{J,2}}_{v}(t)\right\\}\xrightarrow[\hskip
14.22636pt]{d}\begin{dcases}\mathbb{E}^{{}^{J,2}}_{v}(t)&\text{if
}\widehat{f}^{{}^{\hskip
1.42262ptJ,2}}_{v}(t)=\widehat{C}_{n}(t,v)+\widehat{C}_{n}(1-v,t)-v\\\
\widehat{\mathbb{E}}^{{}^{J,2}}_{v}(t)&\text{if }\widehat{f}^{{}^{\hskip
1.42262ptJ,2}}_{v}(t)=\widehat{D}_{n}(t,v)+\widehat{D}_{n}(1-v,t)-v\end{dcases}$

in $\ell^{\infty}([0,1])$ as $n\rightarrow\infty$, where
$\mathbb{E}^{{}^{J,2}}_{v}$ and $\widehat{\mathbb{E}}^{{}^{J,2}}_{v}$ are two
centered Gaussian random fields.

The proof of this Proposition can be found in Appendix A.3.

###  2.3 Visualization

The visualization of test functions can be achieved using functional boxplots
(Sun and Genton, 2011). To construct the functional boxplot of
$\widehat{f}^{{}^{\hskip 1.42262ptS}}_{v_{1}},\ldots,\widehat{f}^{{}^{\hskip
1.42262ptS}}_{v_{m}},\widehat{g}^{{}^{\hskip
1.42262ptS}}_{w_{1}},\ldots,\widehat{g}^{{}^{\hskip 1.42262ptS}}_{w_{m}}$, we
need to discretize the interval $[0,1]$ into an evenly spaced range of $p$
points for some properly chosen $p\in\mathbb{N}$ and then evaluate the test
functions $\widehat{f}^{{}^{\hskip
1.42262ptS}}_{v_{1}},\ldots,\widehat{f}^{{}^{\hskip
1.42262ptS}}_{v_{m}},\widehat{g}^{{}^{\hskip
1.13809ptS}}_{w_{1}},\ldots,\widehat{g}^{{}^{\hskip 1.13809ptS}}_{w_{m}}$ over
them. We denote the range of $p$ points by $\\{0\leq t_{1},\ldots,t_{p}\leq
1\\}$. According to Proposition 1, for every $j\in\\{1,\ldots,m\\}$, the joint
distribution of the random vector $\big{(}\widehat{f}^{{}^{\hskip
1.42262ptS}}_{v_{j}}(t_{1})-f^{{}^{S}}_{v_{j}}(t_{1}),\ldots,\widehat{f}^{{}^{\hskip
1.42262ptS}}_{v_{j}}(t_{p})-f^{{}^{S}}_{v_{j}}(t_{p})\big{)}^{\top}$ is close
to a multivariate normal distribution with mean zero when the sample size $n$
is sufficiently large, and so is the joint distribution of the random vector
$\big{(}\widehat{g}^{{}^{\hskip
1.13809ptS}}_{w_{j}}(t_{1})-g^{{}^{S}}_{w_{j}}(t_{1}),\ldots,\widehat{g}^{{}^{\hskip
1.13809ptS}}_{w_{j}}(t_{p})-g^{{}^{S}}_{w_{j}}(t_{p})\big{)}^{\top}$. Thus,
the functional boxplot should be fairly centered around zero (even though its
shape should not be expected as a horizontal band since the variance at each
point $t_{k},k\in\\{1,\ldots,p\\},$ can vary based on the underlying copula
$C$). If the underlying copula $C$ is reflection symmetric, it implies the
equalities $f^{{}^{S}}_{v_{1}}\equiv\cdots\equiv f^{{}^{S}}_{v_{m}}\equiv
g^{{}^{S}}_{w_{1}}\equiv\cdots\equiv g^{{}^{S}}_{w_{m}}\equiv 0$. Otherwise,
the functional boxplot should be of an irregular shape that is visually
deviated from zero. In the special case when $v_{i}\approx w_{i}$ for a great
proportion of $i\in\\{1,\ldots,n\\}$, the functional boxplot may form an
irregular envelope that is almost symmetric with respect to zero.

To assess radial symmetry, we analyze the functional boxplots of
$\widehat{f}^{{}^{\hskip 1.42262ptR}}_{v_{1}},\ldots,\widehat{f}^{{}^{\hskip
1.42262ptR}}_{v_{m}},\widehat{g}^{{}^{\hskip
1.13809ptR}}_{w_{1}},\ldots,\widehat{g}^{{}^{\hskip 1.13809ptR}}_{w_{m}}$.
According to Proposition 2, the functional boxplots should be fairly centered
around zero if and only if the underlying copula $C$ is indeed radial
symmetric. For joint symmetry, two types of test functions are defined,
resulting in two functional boxplots per copula. Proposition 3 states that
both functional boxplots should be centered around zero if the underlying
copula is jointly symmetric.

To stress the interpretation, we demonstrate the functional boxplots based on
various copulas models in Figure 1. The gradation of the colors used in the
plots manifests the density of functional data: the darker the colors are, the
more functional curves are located in place. The corresponding structure(s) of
each copula is flagged by the abbreviation(s) in bold: S (reflection
symmetry), R (radial symmetry), J (joint symmetry) and AS (asymmetry) in the
title of their respective functional boxplots.

Figure 1: Functional boxplots for the visualization of copula symmetries. The
parameter(s) of each copula is chosen to make its Kendall’s tau equal/close to
$0.5$. Especially, the Marshall-Olkin copula takes the parameters
$(0.55,0.85)$, and the Tawn copula, regardless of its types, takes the
parameters $(4.28,0.60)$. The corresponding test functions are constructed
with $n=1000,m=250$ and $p=100$, and the values of $v_{j},w_{j}$ for all
$j\in\\{1,\ldots,m\\}$ are randomly chosen from the unit interval $[0,1]$.
Note that the plots of copulas that inherit the specified symmetries are
featured by green-centered regions, whereas those who disobey the structures
are featured by red ones.

###  2.4 Rank-based hypothesis testing procedure

In Section 2.3, we explored the visualization of copula structures present in
a given sample or dataset. We discussed the relevant asymptotic results and
demonstrated how functional boxplots of selected test functions can provide
intuitive indications of various symmetries. To thoroughly investigate and
quantify these symmetries, we introduce a one-sample ranked-based hypothesis
testing procedure. This procedure is a modification of the methods proposed by
Huang and Sun (2019) and Huang et al. (2023) for testing the separability and
symmetry of univariate and multivariate covariance functions. Our adapted
approach allows us to assess the presence of symmetries in copula structures
in a robust and statistically rigorous manner. Both of these methods can be
considered as adaptations of the two-sample rank-based test proposed by López-
Pintado and Romo (2009). The original test is designed to determine whether
two sets of functional data are derived from the same distribution. In a
similar manner, we modify this test to examine reflection symmetry as the
initial step. Subsequently, by making specific adjustments, the procedure can
be extended to test radial and joint symmetry.

Suppose that we have the observations $\bm{U}_{i}$’s or the pseudo-
observations $\widehat{\bm{U}}_{i}$’s for $i=1,\ldots,n$. As expected, for
$v\in[0,1]$ the null $\mathcal{H}_{0}$ and the alternative $\mathcal{H}_{a}$
are

  * $\bullet\hskip 5.69054pt\mathcal{H}_{0}\hskip 5.69054pt$:

$f^{{}^{S}}_{v}=g^{{}^{S}}_{v}=0\leavevmode\nobreak\ $;

  * $\bullet\hskip 5.69054pt\mathcal{H}_{a}\hskip 5.69054pt$:

$f^{{}^{S}}_{v}=g^{{}^{S}}_{v}\neq 0$.

The details of the procedure are demonstrated as follows:

  1. Step 1:

Estimate the values of the reflection symmetry test functions
$\widehat{f}^{{}^{\hskip 1.42262ptS}}_{v_{1}},\ldots,\widehat{f}^{{}^{\hskip
1.42262ptS}}_{v_{m}},\widehat{g}^{{}^{\hskip
1.13809ptS}}_{w_{1}},\ldots,\widehat{g}^{{}^{\hskip 1.13809ptS}}_{w_{m}}$ over
the evenly spaced points $\\{0\leq t_{1},\ldots,t_{p}\leq 1\\}$ as in Table 1
by $\bm{U}_{i}$’s, or $\widehat{\bm{U}}_{i}$’s, where for any
$j\in\\{1,\ldots,m\\}$, $v_{j}$ and $w_{j}$ are randomly generated from the
unit interval $[0,1]$.

  2. Step 2:

Simulate from a reflection symmetric copula to obtain a set of $n$
observations, $\bm{V}^{{}^{\mathcal{H}_{0}}}_{i},\hskip
2.84526pti\in\\{1,\ldots,n\\}$. Further details on how to simulate the
observations $\bm{V}^{{}^{\mathcal{H}_{0}}}_{i}$ are described in Section
2.4.1.

  3. Step 3:

Estimate the values of the reflection symmetry test functions
$\widehat{f}^{{}^{\hskip
2.84526pt\mathcal{H}_{0}}}_{v_{1}},\ldots,\widehat{f}^{{}^{\hskip
2.84526pt\mathcal{H}_{0}}}_{v_{m_{0}}},\widehat{g}^{{}^{\hskip
2.84526pt\mathcal{H}_{0}}}_{w_{1}},\ldots,\widehat{g}^{{}^{\hskip
2.84526pt\mathcal{H}_{0}}}_{w_{m_{0}}}$ over the evenly spaced points
$\\{0\leq t_{1},\ldots,t_{p}\leq 1\\}$ as in Table 1 by
$\bm{V}^{\mathcal{H}_{0}}_{i}$’s, where for any
$j^{0}\in\\{1,\ldots,m_{0}\\}$, $v_{j^{0}}$ and $w_{j^{0}}$ are randomly
generated from the unit interval $[0,1]$.

  4. Step 4:

Combine the values of $\widehat{f}^{{}^{\hskip
1.42262ptS}}_{v_{1}},\ldots,\widehat{f}^{{}^{\hskip
1.42262ptS}}_{v_{m}},\widehat{g}^{{}^{\hskip
1.13809ptS}}_{w_{1}},\ldots,\widehat{g}^{{}^{\hskip 1.13809ptS}}_{w_{m}}$ with
those of $\widehat{f}^{{}^{\hskip
2.84526pt\mathcal{H}_{0}}}_{v_{1}},\ldots,\widehat{f}^{{}^{\hskip
2.84526pt\mathcal{H}_{0}}}_{v_{m_{0}}},\widehat{g}^{{}^{\hskip
2.84526pt\mathcal{H}_{0}}}_{w_{1}},\ldots,\widehat{g}^{{}^{\hskip
2.84526pt\mathcal{H}_{0}}}_{w_{m_{0}}}$ to form a discretized functional data
set of size $2(m+m_{0})$. Compute their modified band depths (see e.g., López-
Pintado and Romo, 2009; Sun et al., 2012).

  5. Step 5:

Rank the $2(m+m_{0})$ test functions according to their depth values. In case
of any ties, we assign distinct ordinal numbers at random to the test
functions that compare equal in terms of the depth values. Suppose that
$\widehat{f}^{{}^{\hskip 1.42262ptS}}_{v_{1}},\ldots,\widehat{f}^{{}^{\hskip
1.42262ptS}}_{v_{m}},\widehat{g}^{{}^{\hskip
1.13809ptS}}_{w_{1}},\ldots,\widehat{g}^{{}^{\hskip 1.13809ptS}}_{w_{m}}$ are
associated with the ranks $r_{1},\ldots,r_{2m_{0}}$. Define the test statistic
$W=\sum_{i=1}^{2m_{0}}r_{i}$.

The null hypothesis $\mathcal{H}_{0}$ is rejected when $W$ is significantly
small because it means that the test functions $\widehat{f}^{{}^{\hskip
1.42262ptS}}_{v_{1}},\ldots,\widehat{f}^{{}^{\hskip
1.42262ptS}}_{v_{m}},\widehat{g}^{{}^{\hskip
1.13809ptS}}_{w_{1}},\ldots,\widehat{g}^{{}^{\hskip 1.13809ptS}}_{w_{m}}$ are
more deviated from zero. The definition of the test statistic $W$ here takes
the essence of the Wilcoxon, or equivalently Mann-Whitney, test statistic.
Hence, one can also deem the proposed hypothesis testing as a modification of
the two-sample Wilcoxon rank-sum test for one functional data set. The null
distribution of $W$ is estimated by $N_{b}$ bootstrap samples of size $n$.
More specifically, we generate $N_{b}$ samples from the reflection symmetry
copula in Step 2. For the $b$-th sample, $b\in\\{1,\ldots,N_{b}\\}$, we regard
it as a set of $\bm{U}_{i}$’s and follow the above procedure to calculate the
corresponding test statistic $W_{b}$. Eventually, the null distribution of $W$
is approximated by all of these test statistics: $W_{1},\ldots,W_{N_{b}}$. For
the details on how to carry out such a one-sample bootstrapping method in
hypothesis testing, we refer to Section 16.4 in Efron and Tibshirani (1993).

Some features of the procedure are worth further discussion. First, the number
of observations simulated from a reflection symmetry copula needs to be the
same as the original sample size $n$. Otherwise, even if the test functions
$\widehat{f}^{{}^{\hskip 1.42262ptS}}_{v_{1}},\ldots,\widehat{f}^{{}^{\hskip
1.42262ptS}}_{v_{m}},\widehat{g}^{{}^{\hskip
1.13809ptS}}_{w_{1}},\ldots,\widehat{g}^{{}^{\hskip 1.13809ptS}}_{w_{m}}$ came
from a reflection symmetry copula, they would still be more centered/deviated
with respect to zero, thereby having greater depth values, compared to
$\widehat{f}^{{}^{\hskip
2.84526pt\mathcal{H}_{0}}}_{v_{1}},\ldots,\widehat{f}^{{}^{\hskip
2.84526pt\mathcal{H}_{0}}}_{v_{m_{0}}},\widehat{g}^{{}^{\hskip
2.84526pt\mathcal{H}_{0}}}_{w_{1}},\ldots,\widehat{g}^{{}^{\hskip
2.84526pt\mathcal{H}_{0}}}_{w_{m_{0}}}$. This is because larger or smaller
sample sizes make empirical copulas better or worse approximations of the
underlying true copulas.

Second, the values of $p$, $m$, and $m_{0}$ in Steps 1 and 3 are
hyperparameters that one can tune at will. Nonetheless, it is observed that
setting $p=100$ is sufficient, while $m$ and $m_{0}$ have better to be large
to guarantee the ideal empirical size and power of the test. Particularly, we
notice that having $m$ and $m_{0}$ greater than $n$ yields better
performances. Once both $m$ and $m_{0}$ satisfy the property, their values
tend to have little effect on its performance.

Third, the most notable difference between our procedure and those used in
López-Pintado and Romo (2009), Huang and Sun (2019) and Huang et al. (2023)
lies in the absence of a reference data set. The ideas in their approaches
originate with the hypothesis testing procedure designed by Liu and Singh
(1993) that performs the detection of (dis)similarity in two multivariate
distributions using quality index. Particularly, they introduced the treatment
with a reference data set to identify the change in population locations. In
the context of functional data, it describes the scenario when the two samples
of functional curves appear to form two separated bands, each of which
comprises only curves from one group as shown in Figure 2. However, it should
not be a concern in our testing procedure thanks to the introduction of
$\widehat{g}^{{}^{\hskip 1.13809ptS}}_{w_{1}},\ldots,\widehat{g}^{{}^{\hskip
1.13809ptS}}_{w_{m}}$ which are rough reflections of the test functions
$\widehat{f}^{{}^{\hskip 1.13809ptS}}_{v_{1}},\ldots,\widehat{f}^{{}^{\hskip
1.13809ptS}}_{v_{m}}$ with respect to zero. As one may see later, we actually
generate the observations $\bm{V}^{\mathcal{H}_{0}}_{i},i\in\\{1,\ldots,n\\}$
from a mixture ($M$) of empirical copulas in Step 2 under the assumption that
$M$ is close to a truly reflection symmetry copula. Then, if we had to
simulate a reference data set, the empirical copula of the reference data set
might be more symmetric than that of the observations
$\bm{V}^{\mathcal{H}_{0}}_{i},i\in\\{1,\ldots,n\\}$ when the sample size $n$
is not sufficiently large. This could lead to a not-so-small test statistic in
the approach of Huang and Sun (2019) or Huang et al. (2023) as the test
functions constructed from the reference data set might tend to be slightly
more centered around zero. Certainly, the absence of a reference data set also
saves memory storage.

Figure 2:  In the context of functional data, the approach of Liu and Singh
(1993) can be interpreted as the scenario where two samples of functional
curves exhibit a distinct pattern. Specifically, the curves from each group
form separate bands, with little to no overlap between the two groups.

####  2.4.1 Simulation under $\mathcal{H}_{0}$

Lastly, it is important to address the choice of the reflection symmetric
copula in Step 2 and explain how to simulate from it. In order to maintain the
nonparametric nature of the procedure, it is crucial to avoid selecting any
parametric copula. In lieu, we propose to construct the estimated reflection
symmetric copula $\widehat{C}^{S}$ given, for all $(u,v)$, by either

$\widehat{C}^{S}(u,v)=\frac{1}{2}\Big{(}\widehat{C}_{n}(u,v)+\widehat{C}_{n}(v,u)\Big{)}$

or

$\widehat{C}^{S}(u,v)=\frac{1}{2}\Big{(}\widehat{D}_{n}(u,v)+\widehat{D}_{n}(v,u)\Big{)},$

depending on the types of data set we have.

######  Proposition 4.

Given any bivariate copula $C$, the copula $\tilde{C}$ defined, for all
$(u,v)\in[0,1]^{2}$, by

$\tilde{C}(u,v)=\frac{1}{2}\Big{(}C(u,v)+C(v,u)\Big{)}$

is a reflection symmetric copula.

We prove this result in the Appendix A.4.

As a mixture of two empirical copulas, one should have no difficulty
simulating from $\widehat{C}^{S}$. Note that $\widehat{C}^{S}$ is not really a
copula itself, but it serves as an appropriate estimation of the reflection
symmetric copula $\tilde{C}$ (Rüschendorf, 1976). Apart from the intuition
suggested by Proposition 4, Figure 3 provides a visual assurance for the
simulation method.

Figure 3: Rank plots for the different copula models considered in Section 2.3

By closely following the aforementioned procedure and making appropriate
adjustments, one can conduct tests for radial and joint symmetries.
Specifically, in Step 2, a copula that exhibits radial or joint symmetry needs
to be constructed. Table 2 provides a summary of the proposed estimated
copulas for this purpose, and the following Propositions provide justification
for these choices. To ensure clarity in our notation, we denote the survival
copula of any copula $C$ as $C^{\ast}$. Proofs of the Propositions 5 and 6 can
be found in the Appendices A.5 and A.6, respectively.

######  Proposition 5.

Given any bivariate copula $C$, the copula $\tilde{C}$ defined, for all
$(u,v)\in[0,1]^{2}$, by

$\tilde{C}(u,v)=\frac{1}{2}\Big{(}C(u,v)+C^{\ast}(u,v)\Big{)}$

is a radially symmetric copula.

######  Proposition 6.

Given any copula $C$, the copula $\tilde{C}$ defined, for all
$(u,v)\in[0,1]^{2}$, by

$\tilde{C}(u,v)=\frac{1}{4}\Big{(}C(u,v)+u-C(u,1-v)+v-C(1-u,v)+C^{\ast}(u,v)\Big{)}$

is a jointly symmetric copula.

Type | Access | Simulated Distribution
---|---|---
S | $(U_{1},U_{2})$ | $\widehat{C}^{S}(u,v)=\frac{1}{2}\Big{(}\widehat{C}_{n}(u,v)+\widehat{C}_{n}(v,u)\Big{)}$
$(X_{1},X_{2})$ | $\widehat{C}^{S}(u,v)=\frac{1}{2}\Big{(}\widehat{D}_{n}(u,v)+\widehat{D}_{n}(v,u)\Big{)}$
R | $(U_{1},U_{2})$ | $\widehat{C}^{R}(u,v)=\frac{1}{2}\Big{(}\widehat{C}_{n}(u,v)+\widehat{C}^{\ast}_{n}(u,v)\Big{)}$
$(X_{1},X_{2})$ | $\widehat{C}^{R}(u,v)=\frac{1}{2}\Big{(}\widehat{D}_{n}(u,v)+\widehat{D}^{\ast}_{n}(u,v)\Big{)}$
J | $(U_{1},U_{2})$ | $\widehat{C}^{{J}}(u,v)=\hskip 2.84526pt\tfrac{1}{4}\Big{(}\widehat{C}_{n}(u,v)+u-\widehat{C}_{n}(u,1-v)+v-\widehat{C}_{n}(1-u,v)+\widehat{C}^{\ast}_{n}(u,v)\Big{)}$
$(X_{1},X_{2})$ | $\widehat{C}^{{J}}(u,v)=\hskip 2.84526pt\tfrac{1}{4}\Big{(}\widehat{D}_{n}(u,v)+u-\widehat{D}_{n}(u,1-v)+v-\widehat{D}_{n}(1-u,v)+\widehat{D}^{\ast}_{n}(u,v)\Big{)}$
Table 2: Definitions of estimated copulas constructed to test different types
of symmetry. $\widehat{C}^{\ast}_{n}$ and $\widehat{D}^{\ast}_{n}$ denote the
estimator for the survival copulas associated with the underlying copula.

##  3 Simulation Study

We performed simulations to evaluate the effectiveness of our testing
procedures for the three copula structures described in Section 2.1. In each
simulation, we estimated 1200 test functions, and the tests were conducted at
a nominal level of $5\%$. We utilized $N_{b}=1000$ bootstrap samples for our
analysis. To assess the performance of our approach for testing reflection
symmetry, we compared it to the tests proposed by Genest et al. (2012) and Li
and Genton (2013). The results, including the sizes and powers of the test for
reflection symmetry, are summarized in Table 3. To evaluate the power of our
tests, we introduce asymmetry to the copula models using Khoudraji’s device
(Khoudraji, 1995). Specifically, we used an asymmetric version of a copula
$C(u,v)$, defined for $(u,v)\in[0,1]^{2}$, given by:

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

where $\delta\in(0,1)$. Previous studies (Genest et al., 2012) have shown that
Khoudraji’s device introduces minimal asymmetry when Kendall’s $\tau\leq 0.5$.
The maximum level of asymmetry is typically observed around $\delta=0.5$. The
R codes for implementing the visualization and hypotheses testing procedures
are available at https://github.com/cfjimenezv07/Visualization-and-Assessment-
of-Copula-Symmetry.

| Clayton | Gaussian | Gumbel
---|---|---|---
| $n$ | $n$ | $n$
$\delta$ | $\tau$ |  | 100 |  250 |  500 |  1000 | 100 |  250 |  500 |  1000 | 100 |  250 |  500 |  1000
0 | 1/4 |  E Z I S | 0.044 | 0.048 | 0.059 | 0.062 | 0.053 | 0.052 | 0.046 | 0.066 | 0.050 | 0.063 | 0.059 | 0.063
1/2 | 0.021 | 0.043 | 0.051 | 0.053 | 0.031 | 0.039 | 0.048 | 0.064 | 0.038 | 0.044 | 0.055 | 0.053
3/4 | 0.007 | 0.002 | 0.007 | 0.007 | 0.001 | 0.002 | 0.003 | 0.004 | 0.007 | 0.008 | 0.009 | 0.019
0 | $0^{\ast}$ | 0.045 | 0.060 | 0.059 | 0.060 |  |  |  |  |  |  |  |
1/4 | 0.5 |  R E W O P | 0.121 | 0.297 | 0.547 | 0.796 | 0.089 | 0.267 | 0.552 | 0.850 | 0.119 | 0.312 | 0.609 | 0.903
0.7 | 0.374 | 0.814 | 0.981 | 0.999 | 0.336 | 0.860 | 0.995 | 1.000 | 0.365 | 0.886 | 0.994 | 1.000
0.9 | 0.675 | 0.977 | 0.996 | 1.000 | 0.740 | 0.987 | 0.998 | 1.000 | 0.728 | 0.990 | 1.000 | 1.000
1/2 | 0.5 | 0.149 | 0.358 | 0.630 | 0.872 | 0.187 | 0.511 | 0.826 | 0.990 | 0.264 | 0.690 | 0.929 | 0.998
0.7 | 0.483 | 0.926 | 0.998 | 1.000 | 0.664 | 0.986 | 1.000 | 1.000 | 0.736 | 0.992 | 1.000 | 1.000
0.9 | 0.908 | 1.000 | 1.000 | 1.000 | 0.951 | 1.000 | 1.000 | 1.000 | 0.940 | 1.000 | 1.000 | 1.000
3/4 | 0.5 | 0.087 | 0.210 | 0.334 | 0.557 | 0.178 | 0.450 | 0.743 | 0.959 | 0.301 | 0.656 | 0.930 | 0.995
0.7 | 0.254 | 0.629 | 0.897 | 0.996 | 0.500 | 0.929 | 0.999 | 1.000 | 0.605 | 0.957 | 0.998 | 1.000
0.9 | 0.662 | 0.984 | 1.000 | 1.000 | 0.761 | 0.999 | 1.000 | 1.000 | 0.768 | 0.985 | 1.000 | 1.000

Table 3: Empirical sizes and powers of the test of symmetry in the setting as
Genest et al. (2012) and Li and Genton (2013). The sizes for the independent
copula are $\delta=\tau=0$.

Our results indicate that under small and moderate $\tau$, the sizes converge
to the nominal value as the sample size increases, but with larger $\tau$, the
sizes are somewhat below the nominal level. The powers of the tests increase
as the sample size increases. When comparing our results to Table 3 in Genest
et al. (2012), their approach generally achieves better powers for a small
sample size of $n=250$. In contrast, our approach demonstrates significantly
improved powers as the sample size increases. This is expected as our rank-
based test relies on the asymptotic distribution of the test functions and the
empirical copula process.

When comparing our simulation results to Table 1 in Li and Genton (2013), we
find that our results align with most of the reported powers and sizes.
However, our approach achieves considerably higher powers even at smaller
sample sizes, particularly for intermediate values of $\tau$ where the maximum
asymmetry is expected.

For the tests of radial and joint symmetry, we selected five commonly used
copulas. The sizes and powers of both testing procedures are presented in
Table 4. To evaluate the performance of our approach in testing radial
symmetry, we compared it to the methods proposed by Li and Genton (2013) and
Genest and Nešlehová (2014). Additionally, we compared our approach for
testing joint symmetry to the methods reported by Li and Genton (2013).

| Radial | Joint
---|---|---
| $\tau$ | $n=$ | $100$ | $250$ | $500$ | $1000$ | $n=$ | $100$ | $250$ | $500$ | $1000$
$\Pi$ | $0$ |  E Z I S | 0.060 | 0.073 | 0.065 | 0.060 | SIZE | 0.030 | 0.029 | 0.024 | 0.048
Frank | 1/4 | 0.054 | 0.060 | 0.059 | 0.054 |  R E W O P | 0.685 | 0.963 | 0.997 | 1.000
1/2 | 0.050 | 0.064 | 0.058 | 0.059 | 0.968 | 1.000 | 1.000 | 1.000
3/4 | 0.016 | 0.041 | 0.037 | 0.046 | 0.975 | 0.999 | 1.000 | 1.000
Gaussian | 1/4 | 0.062 | 0.060 | 0.047 | 0.059 | 0.725 | 0.984 | 1.000 | 1.000
1/2 | 0.041 | 0.041 | 0.054 | 0.054 | 0.976 | 0.999 | 1.000 | 1.000
3/4 | 0.009 | 0.022 | 0.014 | 0.031 | 0.973 | 1.000 | 1.000 | 1.000
Clayton | 1/4 |  R E W O P | 0.228 | 0.491 | 0.830 | 0.985 | 0.706 | 0.957 | 0.999 | 1.000
1/2 | 0.516 | 0.833 | 0.985 | 1.000 | 0.966 | 0.999 | 1.000 | 1.000
3/4 | 0.291 | 0.512 | 0.912 | 0.996 | 0.976 | 1.000 | 1.000 | 1.000
Gumbel | 1/4 | 0.110 | 0.333 | 0.510 | 0.695 | 0.696 | 0.974 | 0.998 | 1.000
1/2 | 0.166 | 0.486 | 0.723 | 0.909 | 0.974 | 1.000 | 1.000 | 1.000
3/4 | 0.042 | 0.322 | 0.580 | 0.815 | 0.978 | 0.999 | 1.000 | 1.000

Table 4: Sizes and powers of the test of radial and joint symmetry in the
setting as Li and Genton (2013) and Genest and Nešlehová (2014). One bootstrap
sample is used to estimate the distribution of the test statistic $W$ under
the null.

The sizes of both tests closely align with the nominal level of $5\%$ for
values of $\tau$ equal to $1/4$ and $1/2$. However, for larger values of
$\tau$, the sizes are achieved at larger sample sizes. Specifically, in the
case of the radial symmetry test, our results are consistent with those
presented in Table 2 of Li and Genton (2013), and once again our testing
approach achieves the nominal levels at smaller sample sizes.

In comparison to the results presented in Table 1 of Genest and Nešlehová
(2014) for the Frank and Gaussian copula models they considered, our approach
achieves sizes that are closer to the nominal level. Although their
performance improves for larger sample sizes, our approach outperforms them
even in scenarios with larger sample sizes.

Regarding the joint symmetry copula structure, our achieved powers are
significantly higher compared to the powers obtained for the radial symmetry
property, similar results are presented in Table 2 of Li and Genton (2013).

##  4 Data Applications

To illustrate the procedures herein, we apply our visualization and hypothesis
testing methodology to two real-world data sets. The Nutritional Habits Survey
Data is in Section 4.1, and the wind speed dataset in Saudi Arabia is in
Section 4.2.

###  4.1 Nutritional Habits Survey Data

The dataset used here comes from a survey conducted by the U.S. Department of
Agriculture in 1985. The survey aimed to investigate the dietary habits of 737
women aged between 25 and 50 years. Specifically, the survey collected daily
intake measurements of five variables: calcium (mg), iron (mg), protein (g),
vitamin A (mg), and vitamin C (mg).

In previous analyses, Genest et al. (2012) employed a Cramér-von Mises
statistic to evaluate the bivariate reflection symmetry of the pairwise
copulas. Similarly, Li and Genton (2013) utilized a nonparametric method based
on the asymptotic distribution of the empirical copula process to assess
reflection symmetry for this dataset. In our study, we conducted a test using
our rank-based method and obtained corresponding p-values, which are presented
in the top-right corner of each subplot in Figure 4.

The pattern of p-values obtained in our study exhibits similarities to the
findings reported by Genest et al. (2012) and Li and Genton (2013), leading to
similar conclusions for some pairs. However, there are notable differences.
Unlike Genest et al. (2012), our test does not reject the reflection symmetry
copula at a $5\%$ significance level for the pairs (iron, vitamin A), (iron,
vitamin C), and (protein, vitamin A). Conversely, we do reject the reflection
symmetry copula for the pair (protein, vitamin C).

When comparing our results to those of Li and Genton (2013), we find much
closer agreement. The only difference in conclusions arises for the two pairs:
(protein, vitamin C), where we reject the reflection symmetry copula, and
(iron, vitamin A), where we do not reject it. Notably, the pair (iron, vitamin
A) differs from the conclusions of both Genest et al. (2012) and Li and Genton
(2013). However, our proposed visualization method reveals that the majority
of the test functions exhibit high density around the zero level for this
pair.

Figure 4: Visualizations and testing results for copula symmetries over the
$5$ random variables in the nutritional data set.

###  4.2 Wind data

In this study, we analyze a trivariate wind speed dataset obtained from Yip
(2018). The dataset comprises bi-weekly mid-day wind speed measurements
recorded over the period of 2009-2014 at three specific locations near Dumat
Al Jandal, the site of the first wind farm currently under construction in
Saudi Arabia. The dataset consists of $n=156$ trivariate wind speed vectors,
each containing measurements from three positions.

Understanding the dependence structure within these trivariate wind speed
vectors is crucial for assessing how wind patterns impact the electricity
generation of the nearby wind farm. By analyzing this dependence structure, we
can gain insights into the interactions between wind speeds at different
locations and optimize the operation and output of the wind farm.

One specific aspect of interest is evaluating whether a potential copula
model, which quantifies the dependence among bivariate positions, exhibits a
specific symmetry structure. This assessment allows us to examine the symmetry
properties of a potential copula model and determine if it aligns with the
desired symmetry structure. Understanding the symmetry properties can be
essential for accurately modeling and predicting wind speed behavior.

Figure 5 presents the visualization and hypothesis testing results for the
dataset under consideration, focusing on the three copula structures analyzed
in the paper. The results show that joint symmetry is rejected for all pairs
of locations. However, reflection symmetry and radial symmetry are not
rejected by the bivariate dataset formed by positions 2 and 3. This is
indicated by the functional boxplot, which demonstrates functions distributed
widely around zero and the corresponding large p-values. In summary, our
approach suggests using asymmetric copula models to properly capture the
dependence in these wind speed data.

Figure 5: Visualizations and testing results for copula symmetries over the
$3$ locations for the wind data in Saudi Arabia.

##  5 Discussion

In this paper, we have presented a comprehensive framework for visualizing and
testing common assumptions regarding copula structures, such as reflection
symmetry, radial symmetry, and joint symmetry. Our approach utilizes
functional data analysis techniques to construct test functions based on
bivariate copulas at specific discrete points. These test functions
effectively summarize the copula structures and provide valuable insights into
their adherence to specific structures.

To visually represent the copula structures, we employ functional boxplots,
which depict the functional median and variability of the test functions.
These visualizations allow us to assess the departure from zero and gain
insights into the degree to which the copula structures conform to the desired
assumptions.

Additionally, we have introduced a nonparametric testing procedure to evaluate
the significance of deviations from symmetry. Through extensive simulation
studies involving various copula models, we have demonstrated the reliability
and power of our method, particularly for moderate to large sample sizes. The
numerical experiments and data analyses conducted in this study have
consistently shown robust testing results across different datasets, and the
visualization technique has proven useful in extracting preliminary
information directly from the data. It is worth mentioning that our functional
data approach can be extended to test copula properties beyond symmetry as
well.

It is important to note that our testing method relies on the asymptotic
distribution of the estimators of the test functions, which are derived from
empirical copula processes. As a result, the small sample properties of our
test may not be as optimal as certain existing testing methods. The required
sample size for our testing approach can vary depending on the specific copula
structure under examination, but in general, larger sample sizes tend to
enhance the size and power of the test. Thus, increasing the sample size is
recommended to improve the overall performance of the test in terms of
accuracy and sensitivity.

Finally, our visualization and testing techniques were applied to two real-
world datasets: a nutritional habits survey with five variables and wind speed
data from three locations in Saudi Arabia. These applications provided
valuable insights into the underlying structures and patterns within the
datasets, demonstrating the effectiveness of our approach in gaining a better
understanding of the data.

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##  Appendix A Appendix

###  A.1 Proof of Proposition 1

The proof of the proposition is essentially the same as that of Proposition 2
in Genest et al. (2012). They focus on the asymptotic results when the
underlying copula $C$ is indeed reflection symmetric. We firstly consider the
case when $\widehat{f}^{{}^{\hskip
1.42262ptS}}_{v}(t)=\widehat{C}_{n}(t,v)-\widehat{C}_{n}(v,t)$. The map

$\mathcal{T}^{S}_{v}:\ell([0,1]^{2})\rightarrow\ell([0,1]):f\mapsto\big{(}t\mapsto
f(t,v)-f(v,t)\big{)}\\\ $

is a continuous function. Thus, the Continuous Mapping Theorem (van der Vaart
and Wellner (1996), Theorem 1.3.6) and result (5) imply that
$\sqrt{n}\left\\{\widehat{f}^{{}^{\hskip
1.42262ptS}}_{v}(t)-f^{{}^{S}}_{v}(t)\right\\}\xrightarrow[\hskip
14.22636pt]{d}\mathbb{E}^{{}^{S}}_{v}(t):=\mathbb{C}(t,v)-\mathbb{C}(v,t)$ for
all $t\in[0,1]$, where $\mathbb{E}^{{}^{S}}_{v}$ is a centered Gaussian random
field with covariance function given at each $s,t\in[0,1]$ by

$\displaystyle\mathrm{Cov}\\{\mathbb{E}^{{}^{S}}_{v}(t),\hskip
2.84526pt\mathbb{E}^{{}^{S}}_{v}(s)\\}=$ $\displaystyle\hskip
5.69054pt\mathrm{Cov}\\{\mathbb{C}(t,v)-\mathbb{C}(v,t),\hskip
2.84526pt\mathbb{C}(s,v)-\mathbb{C}(v,s)\\}$ $\displaystyle=$
$\displaystyle\hskip 5.69054ptC(t\land s,v)-C(t,v)C(s,v)+C(v,t\land
s)-C(v,t)C(v,s)$ $\displaystyle-C(t\land v,s\land v)+C(t,v)C(v,s)-C(s\land
v,t\land v)+C(v,t)C(s,v).$

Similarly, if $\widehat{f}^{{}^{\hskip
1.42262ptS}}_{v}(t)=\widehat{D}_{n}(t,v)-\widehat{D}_{n}(v,t)$, we have

$\displaystyle\sqrt{n}\left\\{\widehat{f}^{{}^{\hskip
1.42262ptS}}_{v}(t)-f^{{}^{S}}_{v}(t)\right\\}\xrightarrow[\hskip
14.22636pt]{d}\widehat{\mathbb{E}}^{{}^{S}}_{v}(t):=$ $\displaystyle\hskip
5.69054pt\mathbb{C}(t,v)-\dot{C}_{1}(t,v)\mathbb{C}(t,1)-\dot{C}_{2}(t,v)\mathbb{C}(1,v)$
$\displaystyle-\mathbb{C}(v,t)+\dot{C}_{1}(v,t)\mathbb{C}(v,1)+\dot{C}_{2}(v,t)\mathbb{C}(1,t)$

for all $t\in[0,1]$, where $\widehat{\mathbb{E}}^{{}^{S}}_{v}$ is a centered
Gaussian random field. One can easily derive its covariance function, but the
expression is omitted here in view of its intricate closed form.

###  A.2 Proof of Proposition 2

We first consider the case when $\widehat{f}^{{}^{\hskip
1.42262ptR}}_{v}(t)=\widehat{C}_{n}(t,v)-\widehat{C}_{n}(1-t,1-v)+1-t-v$.
Notice that

$\displaystyle\sqrt{n}\left\\{\widehat{f}^{{}^{\hskip
1.42262ptR}}_{v}(t)-f^{{}^{R}}_{v}(t)\right\\}$
$\displaystyle=\sqrt{n}\left\\{\widehat{C}_{n}(t,v)-\widehat{C}_{n}(1-t,1-v)-\big{(}C(t,v)-C(1-t,1-v)\big{)}\right\\}$
$\displaystyle=\sqrt{n}\left\\{\widehat{C}_{n}(t,v)-C(t,v)\right\\}-\sqrt{n}\left\\{\widehat{C}_{n}(1-t,1-v)-C(1-t,1-v)\right\\}$
$\displaystyle=\mathbb{C}_{n}(t,v)-\mathbb{C}_{n}(1-t,1-v).$

Since the map

$\mathcal{T}^{R}_{v}:\ell([0,1]^{2})\rightarrow\ell([0,1]):f\mapsto\big{(}t\mapsto
f(t,v)-f(1-t,1-v)\big{)}\\\ $

is a continuous functional, the Continuous Mapping Theorem and result (5)
imply that $\sqrt{n}\left\\{\widehat{f}^{{}^{\hskip
1.42262ptR}}_{v}(t)-f^{{}^{R}}_{v}(t)\right\\}$ $\xrightarrow[\hskip
14.22636pt]{d}\mathbb{E}^{{}^{R}}_{v}(t):=\mathbb{C}(t,v)-\mathbb{C}(1-t,1-v)$
for all $t\in[0,1]$, where $\mathbb{E}^{{}^{R}}_{v}$ is a centered Gaussian
random field with covariance function given at each $s,t\in[0,1]$ by

$\displaystyle\mathrm{Cov}\\{$ $\displaystyle\mathbb{E}^{{}^{R}}_{v}(t),\hskip
2.84526pt\mathbb{E}^{{}^{R}}_{v}(s)\\}$ $\displaystyle=$ $\displaystyle\hskip
5.69054pt\mathrm{Cov}\\{\mathbb{C}(t,v)-\mathbb{C}(1-t,1-v),\hskip
2.84526pt\mathbb{C}(s,v)-\mathbb{C}(1-s,1-v)\\}$ $\displaystyle=$
$\displaystyle\hskip 5.69054ptC(t\land s,v)-C(t,v)C(s,v)+C(1-t\lor
s,1-v)-C(1-t,1-v)C(1-s,1-v)$
$\displaystyle-C(t\land(1-s),v\land(1-v))+C(t,v)C(1-s,1-v)$
$\displaystyle-C((1-t)\land s,v\land(1-v))+C(s,v)C(1-t,1-v),$

where for all $a,b\in\mathbb{R}$, $a\lor b=\max\\{a,b\\}$. Similarly, if
$\widehat{f}^{{}^{\hskip
1.42262ptR}}_{v}(t)=\widehat{D}_{n}(t,v)-\widehat{D}_{n}(1-t,1-v)+1-t-v$, we
have

$\displaystyle\sqrt{n}$ $\displaystyle\left\\{\widehat{f}^{{}^{\hskip
1.42262ptR}}_{v}(t)-f^{{}^{R}}_{v}(t)\right\\}=\mathbb{D}_{n}(t,v)-\mathbb{D}_{n}(1-t,1-v)$
$\displaystyle\xrightarrow[\hskip 14.22636pt]{d}\hskip
5.69054pt\widehat{\mathbb{E}}^{{}^{R}}_{v}(t):=\mathbb{C}(t,v)-\dot{C}_{1}(t,v)\mathbb{C}(t,1)-\dot{C}_{2}(t,v)\mathbb{C}(1,v)$
$\displaystyle\hskip
71.13188pt-\mathbb{C}(1-t,1-v)+\dot{C}_{1}(1-t,1-v)\mathbb{C}(1-t,1)+\dot{C}_{2}(1-t,1-v)\mathbb{C}(1,1-v)$

for all $t\in[0,1]$, where $\widehat{\mathbb{E}}^{{}^{R}}_{v}$ is a centered
Gaussian random field. One can derive its covariance function, but the
expression is omitted here in view of its intricate closed form.

###  A.3 Proof of Proposition 3

The proof of this Proposition is similar to those of Propositions 1 and 2 but
with a different choice of continuous functionals, and we will only present
the details of the asymptotic result regarding the test function
$\widehat{f}^{{}^{\hskip 1.42262ptS,1}}_{v}$. Firstly, consider the case when
$\widehat{f}^{{}^{\hskip
1.42262ptS,1}}_{v}(t)=\widehat{C}_{n}(t,v)+\widehat{C}_{n}(t,1-v)-t$. Notice
that

$\displaystyle\sqrt{n}\left\\{\widehat{f}^{{}^{\hskip
1.42262ptS,1}}_{v}(t)-f^{{}^{S,1}}_{v}(t)\right\\}$
$\displaystyle=\sqrt{n}\left\\{\widehat{C}_{n}(t,v)+\widehat{C}_{n}(t,1-v)-\big{(}C(t,v)+C(t,1-v)\big{)}\right\\}$
$\displaystyle=\sqrt{n}\left\\{\widehat{C}_{n}(t,v)-C(t,v)\right\\}+\sqrt{n}\left\\{\widehat{C}_{n}(t,1-v)-C(t,1-v)\right\\}$
$\displaystyle=\mathbb{C}_{n}(t,v)+\mathbb{C}_{n}(t,1-v).$

Since the map

$\mathcal{T}^{R}_{v}:\ell([0,1]^{2})\rightarrow\ell([0,1]):f\mapsto\big{(}t\mapsto
f(t,v)+f(t,1-v)\big{)}\\\ $

is a continuous functional, the continuous mapping theorem and result (5)
imply that

$\sqrt{n}\left\\{\widehat{f}^{{}^{\hskip
1.42262ptJ,1}}_{v}(t)-f^{{}^{J,1}}_{v}(t)\right\\}\xrightarrow[\hskip
14.22636pt]{d}\mathbb{E}^{{}^{J,1}}_{v}(t):=\mathbb{C}(t,v)+\mathbb{C}(t,1-v)\leavevmode\nobreak\
\forall t\in[0,1],$

where $\mathbb{E}^{{}^{J,1}}_{v}$ is a centered Gaussian random field with
covariance function given at each $s,t\in[0,1]$ by

$\displaystyle\mathrm{Cov}\\{$
$\displaystyle\mathbb{E}^{{}^{J,1}}_{v}(t),\hskip
2.84526pt\mathbb{E}^{{}^{J,1}}_{v}(s)\\}$ $\displaystyle=$
$\displaystyle\hskip
5.69054pt\mathrm{Cov}\\{\mathbb{C}(t,v)+\mathbb{C}(t,1-v),\hskip
2.84526pt\mathbb{C}(s,v)+\mathbb{C}(s,1-v)\\}$ $\displaystyle=$
$\displaystyle\hskip 5.69054ptC(t\land s,v)+C(t,v)C(s,v)+C(t\land
s,1-v)+C(t,1-v)C(s,1-v)$ $\displaystyle+C(t\land
s,v\land(1-v))+C(t,v)C(s,1-v)+C(t\land s,v\land(1-v))+C(s,v)C(t,1-v),$

where for all $a,b\in\mathbb{R}$, $a\lor b=\max\\{a,b\\}$. Similarly, if
$\widehat{f}^{{}^{\hskip
1.42262ptS,1}}_{v}(t)=\widehat{D}_{n}(t,v)+\widehat{D}_{n}(t,1-v)-t$, we have

$\displaystyle\sqrt{n}$ $\displaystyle\left\\{\widehat{f}^{{}^{\hskip
1.42262ptR}}_{v}(t)-f^{{}^{R}}_{v}(t)\right\\}=\mathbb{D}_{n}(t,v)+\mathbb{D}_{n}(t,1-v)$
$\displaystyle\xrightarrow[\hskip 14.22636pt]{d}\hskip
5.69054pt\widehat{\mathbb{E}}^{{}^{S,1}}_{v}(t):=\mathbb{C}(t,v)-\dot{C}_{1}(t,v)\mathbb{C}(t,1)-\dot{C}_{2}(t,v)\mathbb{C}(1,v)$
$\displaystyle\hskip
71.13188pt-\mathbb{C}(1-t,1-v)+\dot{C}_{1}(1-t,1-v)\mathbb{C}(1-t,1)+\dot{C}_{2}(1-t,1-v)\mathbb{C}(1,1-v)$

for all $t\in[0,1]$, where $\widehat{\mathbb{E}}^{{}^{S,1}}_{v}$ is a centered
Gaussian random field. One can derive its covariance function, but the
expression is omitted here in view of its intricate closed form.

######  Lemma 1.

The average of bivariate copulas is also a bivariate copula.

###### Proof.

Let $n\in\mathbb{N}$ be fixed, and let $C_{1},\ldots,C_{n}$ be $n$ (not
necessarily mutually distinct) bivariate copulas. Define

$\mathcal{C}(u,v)=\frac{1}{n}\sum_{i=1}^{n}C_{i}(u,v),\hskip
14.22636pt\forall(u,v)\in[0,1].$

If $u$ or $v$ is zero, say $u=0$, we have that

$\mathcal{C}(0,v)=\frac{1}{n}\sum_{i=1}^{n}C_{i}(0,v)=\frac{1}{n}\sum_{i=1}^{n}0=0.$

If $u$ or $v$ is $1$, say $u=1$, one can show that

$\mathcal{C}(1,v)=\frac{1}{n}\sum_{i=1}^{n}C_{i}(1,v)=\frac{1}{n}\sum_{i=1}^{n}v=v.$

For any $(a_{1},a_{2}),(b_{1},b_{2})\in[0,1]^{2}$ with $a_{1}\leq b_{1}$ and
$a_{2}\leq b_{2}$, we have

$\sum_{i_{1}=0}^{1}\sum_{i_{2}=0}^{1}(-1)^{i_{1}+i_{2}}\mathcal{C}(x_{1i_{1}},x_{2i_{2}})=\sum_{i_{1}=0}^{1}\sum_{i_{2}=0}^{1}(-1)^{i_{1}+i_{2}}\Big{(}\frac{1}{n}\sum_{j=1}^{n}C_{j}(x_{1i_{1}},x_{2i_{2}})\Big{)}\\\
=\frac{1}{n}\sum_{j=1}^{n}\Big{(}\sum_{i_{1}=0}^{1}\sum_{i_{2}=0}^{1}(-1)^{i_{1}+i_{2}}C_{j}(x_{1i_{1}},x_{2i_{2}})\Big{)}\geq
0,$

where $x_{j0}=a_{j}$ and $x_{j1}=b_{j}$ for all $j\in\\{1,2\\}$. All of this
proves that $\mathcal{C}$ is a copula. Note that the result should be easily
extended to any mixture of bivariate copulas. ∎

###  A.4 Proof of Proposition 4

Assume that $(U,V)$ is a $C$-distributed random vector. Considering the random
vector $(V,U)$, we have

$\mathds{P}(V\leq u,U\leq v)=\mathds{P}(U\leq v,V\leq u)=C(v,u),\hskip
14.22636pt\forall(u,v)\in[0,1].$

It follows from Sklar’s theorem that the bivariate function $\mathcal{C}$
defined, for all $(u,v)\in[0,1]$, by $\mathcal{C}(u,v)=C(v,u)$ is a copula.
Thus, Proposition 4 ensures that $\tilde{C}$ is also a copula. As for the
reflection symmetric structure, it is clear by the associativity of addition.

###  A.5 Proof of Proposition 5

Proposition 4 ensures that $\tilde{C}$ is a bivariate copula. Moreover, one
can easily notice

$\displaystyle\tilde{C}(u,v)-\tilde{C}(1-u,1-v)$
$\displaystyle=\frac{1}{2}\Big{(}C(u,v)+C(1-u,1-v)+u+v-1\Big{)}$
$\displaystyle\hskip
85.35826pt-\frac{1}{2}\Big{(}C(1-u,1-v)+C(u,v)+(1-u)+(1-v)-1\Big{)}$
$\displaystyle=u+v-1,$

which tells us that $\tilde{C}$ is radially symmetric.

###  A.6 Proof of Proposition 6

To justify that $\tilde{C}$ is a copula, it suffices to show that the
bivariate functions $C_{1}$ and $C_{2}$ given, for all $(u,v)\in[0,1]$, by

$C_{1}(u,v)=u-C(u,1-v)\text{ \hskip 5.69054ptand\hskip 5.69054pt
}C_{2}(u,v)=v-C(1-u,v)$

are both copulas. Let $(U,V)$ be a $C$-distributed random vector. Consider the
random vector $(U,1-V)$. For all $(u,v)\in[0,1]$, we have

$\mathds{P}(U\leq u,1-V\leq v)=\mathds{P}(U\leq u,1-V\leq v)\\\
=\mathds{P}(U\leq u)-\mathds{P}(U\leq u,V\leq 1-v)=u-C(u,1-v),\hskip
14.22636pt\forall(u,v)\in[0,1].$

It follows from Sklar’s theorem that $C_{1}$ is a copula; similarly, $C_{2}$
can be shown to be the copula of $(1-u,v)$. Finally, one can check that

$\displaystyle\tilde{C}$ $\displaystyle(u,v)+\tilde{C}(u,1-v)$
$\displaystyle=\frac{1}{4}\Big{(}C(u,v)+u-C(u,1-v)+v-C(1-u,v)+u+v-1+C(1-u,1-v)\Big{)}$
$\displaystyle\hskip
14.22636pt+\frac{1}{4}\Big{(}C(u,1-v)+u-C(u,v)+1-v-C(1-u,1-v)$
$\displaystyle\hskip 284.52756pt+u+1-v-1+C(1-u,v)\Big{)}$
$\displaystyle=\frac{1}{4}(2u+2v-1+2u-2v+1)=u$

and similarly verify $\tilde{C}(u,v)+\tilde{C}(1-u,v)=v$. With all of this, it
can be concluded that $\tilde{C}$ is a jointly symmetric copula.