A Procedure of Perturbation Leading to a New Class of Asymmetric Copulas

Abstract

We propose and analyze a new class of copulas (generated by two univariate functions) that exhibit asymmetric nonlinear dependence, resulting from an antisymmetric perturbation of the independence copula. Within this framework, we construct a parametric subclass of asymmetric copulas for which we derive the analytical expression of Kendall’s tau ($\tau$). We present properties related to asymmetry and dependence measures and provide several examples. This novel feature of our class represents a significant advancement in the context of the most studied perturbed copulas, which have recently been the focus of numerous scientific investigations. In particular, we introduce Liebscher-type constructions and FGM perturbations, resulting in a large general framework for perturbation theory of copulas using elementary mathematical analysis. In addition, we illustrate statements with examples and figures.

1. Introduction

Copulas are mathematical tools and objects that capture and model the dependence structure between random variables by constructing (for instance, bivariate or multivariate) distributions with given marginal distributions. Consider a two-dimensional random vector $(X,Y)$ with joint distribution function $F_{XY}$ and continuous margins $F_X$ and $F_Y$. There exists a copula C such that $F_{XY}(x,y)=C(F_X\left(x\right),F_Y\left(y\right))$.

Here, C represents the distribution function of the vector $(U,V)$, where $U=F_X\left(X\right)$ and $V=F_Y\left(Y\right)$ are uniform random variables on $[0,1]$. Copulas thus allow us to describe the dependence structure between X and Y independently of their marginal distributions. They have gained significant importance and popularity in various fields such as finance, insurance, risk management, and reliability theory (see [1,2,3,4,5]).

Over the years, many copula families and various methods for their construction have been proposed and studied (some works that have inspired our research include [6,7,8,9,10,11,12]).

Our study focuses on a bivariate copula construction principle based on modifying a known copula (particularly the independence copula, denoted by $\Pi$, where $\Pi (u,v)=uv$) by adding a factorial term to its expression. We are thus interested in copulas that can be expressed as follows:

$$C^{\left(P\right)}(u,v)=uv+P(u,v),$$

(1)

where P is a continuous function on the unit square with values in $\mathbb{R}$, called the “perturbation factor”. In this case, the copula C is referred to as a perturbation of $\Pi$ by means of P. P can be interpreted as a disruption of independence (see Durante et al. [6]).

This type of construction underlies several copula families. For example, we may cite the well-known Farlie–Gumbel–Morgenstern FGM family, given by the following:

$$C_{\theta }^{\left(P\right)}(u,v)=uv+\theta uv(1-u)(1-v),\mathrm{for}\mathrm{all}\theta \in [-1,1].$$

(2)

Several extensions of this family, along with their properties, have recently been studied by Saminger-Platz et al. [13]. When the perturbation P is not symmetric, we refer to [12], where a new family of asymmetric copulas has been detailed.

The remainder of this manuscript is organized as follows. We begin by reviewing the definitions, properties, and key results concerning copulas, which are central to our analysis. Next, we describe a particular class of asymmetric copulas based on a slight perturbation of the product copula $\Pi$, while providing necessary and sufficient conditions on the generating functions of this perturbation. Faced with the inability of the FGM copula to model asymmetric relationships, we introduce a generalization through a controlled parametric perturbation, allowing for the simultaneous representation of both symmetric and directional dependence structures. Subsequently, we examine the main dependence property characterizing this copula family, relying on a specific class of weak concordance measures for bivariate copulas recently studied by Mesiar et al. [14], which encompasses most well-known concordance measures in the literature. Finally, we investigate and quantify the asymmetry of our new copula class.

2. Preliminaries

In this contribution, for the sake of simplicity, only bivariate copulas will be considered and treated, and they will simply be referred to as copulas.

Definition 1.

A function $C:I^2\to I$ is called a copula if it satisfies the following two conditions:

$\left(C_1\right)$

0 and 1 are, respectively, its zero element and identity argument; i.e.,

$$C(x,i)=C(i,x)=\left\{\begin{array}{ccc}0,\hfill & if& i=0;\hfill \\ x,\hfill & if& i=1.\hfill \end{array}\right.\mathit{for}\mathit{every}x\in I$$

(3)

$\left(C_2\right)$

It is 2-increasing; i.e., for all $u_1,u_2,v_1,v_2\in I$ where $u_1⩽u_2$ and $v_1⩽v_2$, the volume over the rectangle $R=[u_1,u_2]\times [v_1,v_2]$ is

$$V_c\left(R\right)=C(u_2,v_2)+C(u_1,v_1)-C(u_1,v_2)-C(u_2,v_1)⩾0,$$

(4)

Remark 1.

1. 

Condition $\left(3\right)$ may be compactly written as follows:

$$C(x,i)=C(i,x)=x{\delta }_{i1},\mathit{for}\mathit{every}i\in \{0,1\}.$$

2. 

If C has second derivatives, then the 2-increasing property $\left(4\right)$ is equivalent to see [15].

$${\displaystyle \frac{{\partial }^{2}C(u,v)}{\partial u\partial v}}⩾0,\mathit{for}\mathit{every}(u,v)\in I^2$$

In what follows, we denote by $\mathcal{C}$ the set of all (bivariate) copulas and let $I:=[0,1]$. The set $\mathcal{C}$ possesses many interesting properties, such as being a closed convex subset of the set of functions from the unit square to the unit interval. The Fréchet–Hoeffding bounds W (the countermonotonicity copula) and M (the comonotonicity copula) are, respectively, its smallest and largest elements, given by $W(u,v)=max(u+v-1,0)$ and $M(u,v)=min(u,v)\mathrm{for}\mathrm{every}(u,v)\in I^2$. Furthermore, for every $C\in \mathcal{C}$, C is increasing in each variable—i.e., $C(u,v)⩽C(u^{\prime },v)$ and $C(u,v)⩽C(u,v^{\prime })$ for all $u⩽u^{\prime }$ and $v⩽v^{\prime }$—and is also 1-Lipschitz—i.e., for all $u,u^{\prime },v,v^{\prime }\in I,|C(u,v)-C(u^{\prime },v^{\prime })|⩽|u-u^{\prime }|+|v-v^{\prime }|$. It is also invariant under strictly increasing transformations.

For more details on the theory of copulas, we recommend monographs [16,17].

Each copula can be decomposed into a sum of an absolutely continuous component $A_c$ and a singular component $S_c$ given as follows, respectively:

$$A_c={\int }_0^u{\int }_0^v{\displaystyle \frac{{\partial }^{2}C(x,y)}{\partial x\partial y}}dxdy,\mathrm{and}{S}_c=C-A_c,$$

where, when $C=A_c$ ($i.e.,$$S_c=0$), we say that C is absolutely continuous.

Note that any absolutely continuous copula can be represented as a perturbation of $\Pi$ (see [6]).

Let us now recall the involutive operations on set $\mathcal{F}$ of all two-dimensional random vectors $(X,Y)$ and their connections with respective involutive operations on set $\mathcal{C}$.

• The transformation $(X,Y)\to (X,Y)$ corresponds to the involutive transformation $i:\mathcal{C}\to \mathcal{C},i\left(C\right)(u,v)=C(u,v)$;
• The transformation $(X,Y)\to (Y,X)$ corresponds to the involutive transformation $\pi :\mathcal{C}\to \mathcal{C},\pi \left(C\right)(u,v)=C(v,u)$;
• The transformation $(X,Y)\to (-X,Y)$ corresponds to the involutive transformation ${\varphi }_1:\mathcal{C}\to \mathcal{C},{\varphi }_1\left(C\right)(u,v)=v-C(1-u,v)$;
• The transformation $(X,Y)\to (X,-Y)$ corresponds to the involutive transformation ${\varphi }_2:\mathcal{C}\to \mathcal{C},{\varphi }_2\left(C\right)(u,v)=u-C(u,1-v)$;
• The transformation $(X,Y)\to (-X,-Y)$ corresponds to the involutive transformation $\varphi :\mathcal{C}\to \mathcal{C},\varphi \left(C\right)(u,v)=u+v-1+C(1-u,1-v)$;
• The transformation $(X,Y)\to (-Y,-X)$ corresponds to the involutive transformation $\sigma :\mathcal{C}\to \mathcal{C},\sigma \left(C\right)(u,v)=u+v-1+C(1-v,1-u)$.

Note that $\pi$ is called the transpose, ${\varphi }_1$ is the first flipping, ${\varphi }_2$ is the second flipping, and $\varphi$ is a survival operation.

The analysis of transformations applied to copulas constitutes an active research field (see [18]).

In cases where

• $\pi \left(C\right)(u,v)=C(u,v),\mathrm{for}\mathrm{every}(u,v)\in I^2$, C is called symmetric;
• $\varphi \left(C\right)(u,v)=C(u,v),\mathrm{for}\mathrm{every}(u,v)\in I^2$, C is called radially symmetric.

The latter case was intensively investigated in [19], where invariance of a copula under the transformation $\varphi$ was deeply discussed.

Example 1.

1. 

For all $\theta \in I$, every member $C_{\theta }$ of the FGM family given by (2) is both symmetric and radially symmetric;

2. 

The Fréchet copulas $\lambda M+\lambda W+(1-2\lambda )\Pi ,\lambda \in [0,1/2]$, are also both symmetric and radially symmetric.

Theorem 1.

A copula $C_s$ is symmetric if and only if there exists a copula C such that

$$C_s\left(C\right)(u,v)={\displaystyle \frac{C(u,v)+\pi \left(C\right)(u,v)}{2}},\mathit{for}\mathit{every}(u,v)\in I^2.$$

(5)

It is clear to see that for any symmetric copula S, we have $C_s\left(S\right)=S$. However, what appears more interesting is the inverse problem. This leads us to the fundamental question: given a symmetric copula S, do there exist other asymmetric copulas satisfying the equality $C_s\left(C\right)=S$? In this regard, the authors of [12] have provided an answer to this problem for some notable copulas: $M,W$, and $\Pi$.

Proposition 1.

(i) 

M is the unique copula C satisfying $C_s\left(C\right)=M$;

(ii) 

W is the unique copula C satisfying $C_s\left(C\right)=W$;

(iii) 

There exist numerous copulas C satisfying

$$C_s\left(C\right)=\Pi .$$

(6)

Proof. 

(i)

Let C be a copula such that $C_s\left(C\right)=M$; then $C-M=M-\pi \left(C\right)$. We thus obtain two functions $C-M$ and $M-\pi \left(C\right)$, one negative and the other positive, which are equal. This allows us to conclude that $C-M=0$, hence the uniqueness.

(ii)

The same reasoning as above applies.

(iii)

For this proposition, they only proposed a parametric family of asymmetric copulas satisfying $\left(6\right)$, given by

$C_{\theta }(u,v)=uv+\theta (u-v)min(u,v)min(1-u,1-v),$ for every $(u,v)\in I^2$, with $\theta \in I$. □

Our study proposes a new class of families of copulas satisfying (6), which means the copulas whose symmetrized version $C_s$ is identical to the product copula $\Pi$. The following result can be easily obtained.

Proposition 2.

Let $P:I^2\to \mathbb{R}$ be a continuous function. Then, C defined by $C(u,v)=\Pi (u,v)+P(u,v)$ satisfying (6) is a copula if and only if the following conditions hold:

(i) 

$P(u,1)=P(1,v)=P(u,0)=P(0,v)=0,\mathit{for}\mathit{every}(u,v)\in I^2$

(i.e., $P(u,i)=P(i,v)={\delta }_{i1}{\delta }_{i0},i\in \{0,1\}$);

(ii) 

$P(u,v)=-P(v,u),\mathit{for}\mathit{every}(u,v)\in I^2$;

(iii) 

$V_{\Pi }\left(R\right)+V_P\left(R\right)⩾0,$ for all rectangles $R=[u_1,u_2]\times [v_1,v_2]\subset I^2$.

In other words, the perturbation P must satisfy three requirements: First, it must vanish on the boundaries of $I^2$. Secondly, it must be antisymmetric. Thirdly, the amount of negative mass that it can add to each rectangle of $I^2$ is bounded by the positive mass given by $\Pi$.

Remark 2.

Note that any copula C naturally defines a probability measure on $I^2$. If C takes the form (1), the measure $V_c$ of a rectangle R decomposes into the sum of the measures induced by Π and P.

In what follows, we express the above characterization under specific hypotheses by selecting appropriate functions.

3. New Class of Copulas Based on Antisymmetric Perturbations

3.1. Characterization of the New Class of Copulas

We focus here on copulas whose representation can be formulated as follows:

$$C(u,v)=uv+f\left(u\right)g\left(v\right)-f\left(v\right)g\left(u\right),$$

(7)

where $f,g:I\to \mathbb{R}$ are two non-zero continuous functions. In other words, we have naturally chosen the perturbation $P(u,v)=f\left(u\right)g\left(v\right)-f\left(v\right)g\left(u\right)$.

This form provides a flexible method for introducing new copula families by selecting appropriate functions f and g.

We now need to establish and identify the necessary and sufficient conditions to guarantee that the function defined by (7) is indeed a copula. To achieve this, we begin by stating and proving the following preliminary lemma.

Lemma 1.

Let $f,g:I\to \mathbb{R}$ be two non-zero continuous functions and $P:I\to \mathbb{R}$ a non-zero function defined by $P(u,v)=f\left(u\right)g\left(v\right)-f\left(v\right)g\left(u\right)$. Then, the following two propositions are equivalent:

(i)

$P(i,v)=P(u,i)=0\mathit{for}\mathit{every}i\in \{0,1\},(u,v)\in I^2$;

(ii)

$f\left(0\right)=g\left(0\right)=f\left(1\right)=g\left(1\right)=0$.

Proof. 

The implication $\left(ii\right)\to \left(i\right)$ is obvious, and we therefore focus on the converse.

• From $\left(i\right)$ we have $P(i,v)=0,$ for all $i\in \{0,1\}$, and we thus obtain

$$\begin{array}{cc}\hfill f\left(0\right)g\left(v\right)& =f\left(v\right)g\left(0\right),\forall v\in I;\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill f\left(1\right)g\left(v\right)& =f\left(v\right)g\left(1\right),\forall v\in I.\hfill \end{array}$$

(9)

We assume that at least one of the two functions f and g does not vanish at 0. In this regard, we distinguish two cases.

• If exactly one of the two functions does not vanish at 0, this means that $f\left(0\right)\ne 0$ or $g\left(0\right)\ne 0$.
  For $f\left(0\right)=\alpha$ and $g\left(0\right)=0$ with $\alpha \in {\mathbb{R}}^{*}$, the hypothesis implies that $f\left(0\right)g\left(v\right)=0$ for every $v\in I$, and therefore, $g\left(v\right)=0,$ $v\in I$, which is absurd (since g is considered a continuous non-zero function; i.e., there exists $a\in I$ such that $g\left(a\right)\ne 0$). The same reasoning applies to the other case: $f\left(0\right)=0$ and $g\left(0\right)=\beta$ with $\beta \in {\mathbb{R}}^{*}$.
• If both functions do not vanish at 0, then there exist two non-zero real numbers $\alpha$ and $\beta$ such that $f\left(0\right)=\alpha$ with $g\left(0\right)=\beta$, and substituting the last two equalities into (8), we find that $g\left(v\right)=\gamma f\left(v\right),\mathrm{for}\mathrm{every}v\in I$, where $\gamma =\beta /\alpha$. Hence, $P(u,v)=0$ for every $(u,v)\in I^2$, which is impossible since P is assumed to be non-identically zero.
  Thus, $f\left(0\right)=g\left(0\right)=0$.

• By following the same steps as above for Equation (9), we obtain $f\left(1\right)=g\left(1\right)=0$. In addition, one may deduce the second point in Lemma 1 by considering the functions $\widehat{f}\left(x\right)=f(1-x)$ and $\widehat{g}\left(x\right)=g(1-x)$.
  It follows that $f\left(0\right)=g\left(0\right)=f\left(1\right)=g\left(1\right)=0$. □

We are now in a position to state and prove the central theorem of this section.

Theorem 2.

Let $f,g:I\to \mathbb{R}$ be two non-zero functions, which are continuous on I and differentiable on the interior of I. The function $C:I^2\to \mathbb{R}$ given by (7) is a copula if and only if the following conditions hold:

$\left(1\right)$

$f\left(0\right)=g\left(0\right)=f\left(1\right)=g\left(1\right)=0$;

$\left(2\right)$

f and g are absolutely continuous;

$\left(3\right)$

$f^{\prime }\left(u\right)g^{\prime }\left(v\right)-f^{\prime }\left(v\right)g^{\prime }\left(u\right)⩾-1,$ almost everywhere $\in I^2$.

Proof. 

Note that the equivalence of the first condition of Theorem 2 with the boundary conditions (3) is a direct consequence of Lemma 1. Moreover, the partial derivative of C with respect to u and v (when it exists) is expressed by

$${\displaystyle \frac{{\partial }^{2}C(u,v)}{\partial u\partial v}}=1+f^{\prime }\left(u\right)g^{\prime }\left(v\right)-f^{\prime }\left(v\right)g^{\prime }\left(u\right),a.e.(u,v)\in I^2$$

which entails that ${\displaystyle \frac{{\partial }^{2}C(u,v)}{\partial u\partial v}}⩾0$ (when it exists) is equivalent to $f^{\prime }\left(u\right)g^{\prime }\left(v\right)-f^{\prime }\left(v\right)g^{\prime }\left(u\right)⩾-1,a.e.(u,v)\in I^2$.

Consequently, conditions $\left(2\right)$ and $\left(3\right)$ lead us to conclude that C is 2-increasing.

Conversely, suppose that C is a copula: for each v in I such that $g\left(v\right)\ne 0$ and $f\left(v\right)\ne 0$, the function ${\mathcal{H}}_v\left(t\right)=C(t,v)-\Pi (t,v)=f\left(t\right)g\left(v\right)-f\left(v\right)g\left(t\right)$ formed by the horizontal sections of C and $\Pi$ at $v\in I$ is absolutely continuous for all $t\in I$ (for more details, see Klement et al. [20]).

There thus exists $(v_1,v_2)\in I^2$ such that $P(v_2,v_1)\ne 0$ (P is a non-zero function), where

$$\left\{\begin{array}{ccc}{\mathcal{H}}_{v_1}\left(t\right)\hfill & =& f\left(t\right)g\left(v_1\right)-f\left(v_1\right)g\left(t\right);\hfill \\ {\mathcal{H}}_{v_2}\left(t\right)\hfill & =& f\left(t\right)g\left(v_2\right)-f\left(v_2\right)g\left(t\right).\hfill \end{array}\right.$$

(10)

Therefore,

$$\left\{\begin{array}{ccc}f\left(v_2\right){\mathcal{H}}_{v_1}\left(t\right)\hfill & =& f\left(v_2\right)f\left(t\right)g\left(v_1\right)-f\left(v_2\right)f\left(v_1\right)g\left(t\right);\hfill \\ f\left(v_1\right){\mathcal{H}}_{v_2}\left(t\right)\hfill & =& f\left(v_1\right)f\left(t\right)g\left(v_2\right)-f\left(v_2\right)f\left(v_1\right)g\left(t\right).\hfill \end{array}\right.$$

Hence, $f\left(v_2\right){\mathcal{H}}_{v_1}\left(t\right)-f\left(v_1\right){\mathcal{H}}_{v_2}\left(t\right)=\left(f\left(v_2\right)g\left(v_1\right)-f\left(v_1\right)g\left(v_2\right)\right)f\left(t\right)$.

• By hypothesis ($P(v_2,v_1)\ne 0$), we obtain

$$f\left(t\right)=\alpha {\mathcal{H}}_{v_1}\left(t\right)+\beta {\mathcal{H}}_{v_2}\left(t\right),$$

where

$$\alpha ={\displaystyle \frac{f\left(v_2\right)}{f\left(v_2\right)g\left(v_1\right)-f\left(v_1\right)g\left(v_2\right)}},\mathrm{and}\beta ={\displaystyle \frac{f\left(v_1\right)}{f\left(v_1\right)g\left(v_2\right)-f\left(v_2\right)g\left(v_1\right)}}.$$

Hence, this demonstrates the absolute continuity of f, as a linear combination of two absolutely continuous functions (${\mathcal{H}}_{v_1}$ and ${\mathcal{H}}_{v_2}$).

By multiplying the two equalities in (10) by $g\left(v_2\right)$ and $g\left(v_1\right)$, respectively, a similar reasoning allows us to conclude that g is also absolutely continuous.

The third property of the theorem follows immediately from (4), and the proof is thus completed. □

Let $\mathcal{F}=\{x\in I,f^{\prime }\left(x\right)\mathrm{exists}\}$ and $\mathcal{G}=\{x\in I,g^{\prime }\left(x\right)\mathrm{exists}\}$.

In the case where $\alpha ⩽f^{\prime }\left(x\right)⩽\beta$ and $\gamma ⩽g^{\prime }\left(x\right)⩽\omega$ with $\alpha =inf\{f^{\prime }\left(x\right),x\in \mathcal{F}\}\le 0$, $\beta =sup\{f^{\prime }\left(x\right),x\in \mathcal{F}\}\ge 0$, $\gamma =inf\{g^{\prime }\left(x\right),x\in \mathcal{G}\}\le 0$, and $\omega =sup\{g^{\prime }\left(x\right),x\in \mathcal{G}\}⩾0$. As we have $f^{\prime }\times g^{\prime }\in \left[min\{\alpha \omega ,\beta \gamma \},max\{\alpha \gamma ,\beta \omega \}\right]$, in this case, the third condition of Theorem 2 becomes

$$\mathcal{D}=min\{\alpha \omega ,\beta \gamma \}-max\{\alpha \gamma ,\beta \omega \}⩾-1.$$

(11)

Example 2.

Let $f\left(x\right)={\displaystyle \frac{1}{2}}x(1-x),$ and $g\left(x\right)={\displaystyle \frac{1}{2\pi }}sin\left(\pi x\right)$.

• So, $f^{\prime }\left(x\right)={\displaystyle \frac{1}{2}}(1-2x),$ and $g^{\prime }\left(x\right)={\displaystyle \frac{1}{2}}cos\left(\pi x\right)$.
  Then, $f^{″}\left(x\right)=-1⩽0,$ and $g^{″}\left(x\right)=-{\displaystyle \frac{\pi }{2}}sin\left(\pi x\right)⩽0,\mathit{for}\mathit{all}x\in I$.
  Hence, $f^{\prime }\in [f^{\prime }\left(1\right),f^{\prime }\left(0\right)]=[-{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}],$ and $g^{\prime }\in [g^{\prime }\left(1\right),g^{\prime }\left(0\right)]=[-{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}]$.
  From (11), we then have $\mathcal{D}=-{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2}}=-1⩾-1$.
  It follows that $C(u,v)=uv+{\displaystyle \frac{1}{4\pi }}(u(1-u)sin\left(\pi v\right)-v(1-v)sin\left(\pi u\right))$ defines a copula.

Example 3.

Let $f\left(x\right)=x(1-x^2)$ and $g\left(x\right)={\displaystyle \frac{1}{2\pi }}xcos\left({\displaystyle \frac{\pi }{2}}x\right)$.

• So, $f^{\prime }\left(x\right)=1-3x^2$ and $g^{\prime }\left(x\right)={\displaystyle \frac{1}{2\pi }}cos\left({\displaystyle \frac{\pi }{2}}x\right)-{\displaystyle \frac{1}{4}}xsin\left({\displaystyle \frac{\pi }{2}}x\right)$.
  Then, $f^{″}\left(x\right)=-6x⩽0$ and $g^{″}\left(x\right)=-{\displaystyle \frac{1}{2}}sin\left({\displaystyle \frac{\pi }{2}}x\right)-{\displaystyle \frac{\pi }{8}}xcos\left({\displaystyle \frac{\pi }{2}}x\right)⩽0\mathit{for}\mathit{all}x\in I$.
  Hence, $f^{\prime }\in [f^{\prime }\left(1\right),f^{\prime }\left(0\right)]=[-2,1]$ and $g^{\prime }\in [g^{\prime }\left(1\right),g^{\prime }\left(0\right)]=[-{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{2\pi }}]$.
  It follows that $\mathcal{D}=-{\displaystyle \frac{1}{\pi }}-{\displaystyle \frac{1}{2}}⩾-1$.
  Thus, $C(u,v)=uv(1+{\displaystyle \frac{1}{2\pi }}((1-u^2)cos\left({\displaystyle \frac{\pi }{2}}v\right)-(1-v^2)cos\left({\displaystyle \frac{\pi }{2}}u\right)))$ defines a copula.

3.2. Subclass and Some Examples

This section introduces a parametric subclass of copulas derived from formulation (7), which will be the subject of a more detailed characterization later in our work.

Corollary 1.

Let C be the function given by (7) with $f\left(x\right)=x{g}_{\lambda }\left(x\right)$, where $g_{\lambda }={\lambda }^{{\displaystyle \frac{1}{2}}}g\left(x\right),\mathit{for}\mathit{every}x\in I,\lambda >0$. Then, we obtain

$$C_{\lambda }(u,v)=uv+\lambda (u-v)g\left(u\right)g\left(v\right),$$

(12)

which is a copula if and only if the following conditions hold:

$\left(1\right)$

$g\left(0\right)=g\left(1\right)=0$;

$\left(2\right)$

g is absolutely continuous;

$\left(3\right)$

${\mathcal{V}}_c=g\left(u\right)g^{\prime }\left(v\right)-g\left(v\right)g^{\prime }\left(u\right)+(u-v)g^{\prime }\left(u\right)g^{\prime }\left(v\right)⩾-1/\lambda ,a.e.(u,v)\in I^2$.

Proof. 

The proof is a matter of simple computations of the derivatives for the third condition of Theorem 2. □

Note that the function g plays a role similar to the generating function in Archimedean copulas (see [16]); the inclusion of the coefficient $\lambda$ ensures the validity of the copula construction, and its range depends only on the choice of the function g.

We now propose a few examples of functions g generating parametric families of copulas. One of these examples will be used to illustrate antisymmetry and to evaluate Kendall’s tau in the following sections.

Example 4.

Let m and n be two non-null real numbers such that $mn>0$ (in other words, the parameters m and n have the same sign) and let g be the function defined by $g\left(x\right)=min(mx,{\displaystyle \frac{1-x}{n}}),$ for every $x\in I$.

• If $x⩽{\displaystyle \frac{1}{mn+1}}$, then $mnx⩽1-x$. Hence,

$$\left\{\begin{array}{ccc}mx⩽{\displaystyle \frac{1-x}{n}},\hfill & if& n>0;\hfill \\ mx⩾{\displaystyle \frac{1-x}{n}},\hfill & if& n<0.\hfill \end{array}\right.$$

The function g can then be written in the form

$$g\left(x\right)=\left\{\begin{array}{ccc}mx,\hfill & if& 0⩽x⩽{\displaystyle \frac{1}{mn+1}};\hfill \\ {\displaystyle \frac{1-x}{n}},\hfill & if& {\displaystyle \frac{1}{mn+1}}⩽x⩽1.\hfill \end{array}\right.\mathit{for}\mathit{every}(m,n)\in {\mathbb{R}}_{+}^{*},$$

or

$$g\left(x\right)=\left\{\begin{array}{ccc}{\displaystyle \frac{1-x}{n}},\hfill & if& 0⩽x⩽{\displaystyle \frac{1}{mn+1}};\hfill \\ mx,\hfill & if& {\displaystyle \frac{1}{mn+1}}⩽x⩽1.\hfill \end{array}\right.\mathit{for}\mathit{every}(m,n)\in {\mathbb{R}}_{-}^{*}.$$

Let us consider the case where m and n are positive.

• In this case,

$$g^{\prime }\left(x\right)=\left\{\begin{array}{ccc}m,\hfill & if& 0⩽x⩽{\displaystyle \frac{1}{mn+1}};\hfill \\ {\displaystyle \frac{-1}{n}},\hfill & if& {\displaystyle \frac{1}{mn+1}}⩽x⩽1.\hfill \end{array}\right.\mathit{for}\mathit{every}(m,n)\in {\mathbb{R}}_{+}^{*}.$$

If $0⩽u⩽{\displaystyle \frac{1}{mn+1}}$ and $0⩽v⩽{\displaystyle \frac{1}{mn+1}},$ we have

$${\mathcal{V}}_c=m^{2}u-m^{2}v+m^2(u-v)=m^2(u-v)+m^2(u-v)=2m^2(u-v),$$

$$\left\{\begin{array}{c}0⩽u⩽{\displaystyle \frac{1}{mn+1}};\hfill \\ 0⩽v⩽{\displaystyle \frac{1}{mn+1}}.\hfill \end{array}\right.\mathit{This}\mathit{implies}\mathit{that}{\displaystyle \frac{-1}{mn+1}}⩽u-v⩽{\displaystyle \frac{1}{mn+1}},$$

Hence,

$${\mathcal{V}}_c\in \left[{\displaystyle \frac{-2m^2}{mn+1}},{\displaystyle \frac{2m^2}{mn+1}}\right].$$

If $0⩽u⩽{\displaystyle \frac{1}{mn+1}}$ and ${\displaystyle \frac{1}{mn+1}}⩽v⩽1,$ we have

$${\mathcal{V}}_c={\displaystyle \frac{-m}{n}}u-{\displaystyle \frac{m}{n}}(1-v)-{\displaystyle \frac{m}{n}}(u-v)={\displaystyle \frac{m}{n}}(2v-2u-1),$$

since

$$\left\{\begin{array}{c}{\displaystyle \frac{1}{mn+1}}⩽v⩽1;\hfill \\ 0⩽u⩽{\displaystyle \frac{1}{mn+1}}.\hfill \end{array}\right.\mathit{This}\mathit{implies}\mathit{that}0⩽2v-2u⩽2,$$

Hence,

$${\mathcal{V}}_c\in \left[{\displaystyle \frac{-m}{n}},{\displaystyle \frac{m}{n}}\right].$$

If ${\displaystyle \frac{1}{mn+1}}⩽u⩽1,$ and $0⩽v⩽{\displaystyle \frac{1}{mn+1}}$, we have

$${\mathcal{V}}_c={\displaystyle \frac{m}{n}}(1-u)+{\displaystyle \frac{m}{n}}v-{\displaystyle \frac{m}{n}}(u-v)={\displaystyle \frac{m}{n}}(2v-2u+1),$$

since

$$\left\{\begin{array}{c}0⩽v⩽{\displaystyle \frac{1}{mn+1}};\hfill \\ {\displaystyle \frac{1}{mn+1}}⩽u⩽1.\hfill \end{array}\right.\mathit{This}\mathit{implies}\mathit{that}-2⩽2v-2u⩽0,$$

Hence,

$${\mathcal{V}}_c\in \left[{\displaystyle \frac{-m}{n}},{\displaystyle \frac{m}{n}}\right].$$

If ${\displaystyle \frac{1}{mn+1}}⩽u⩽1,$ and ${\displaystyle \frac{1}{mn+1}}⩽v⩽1$, we have

$${\mathcal{V}}_c=-{\displaystyle \frac{1}{n^2}}(1-u)+{\displaystyle \frac{1}{n^2}}(1-v)+{\displaystyle \frac{1}{n^2}}(u-v)={\displaystyle \frac{1}{n^2}}(u-v)+{\displaystyle \frac{1}{n^2}}(u-v)={\displaystyle \frac{2}{n^2}}(u-v),$$

since

$$\left\{\begin{array}{c}{\displaystyle \frac{1}{mn+1}}⩽u⩽1;\hfill \\ {\displaystyle \frac{1}{mn+1}}⩽v⩽1.\hfill \end{array}\right.\mathit{This}\mathit{implies}\mathit{that}{\displaystyle \frac{-mn}{mn+1}}⩽u-v⩽{\displaystyle \frac{mn}{mn+1}},$$

Hence,

$${\mathcal{V}}_c\in \left[{\displaystyle \frac{-2m}{n(mn+1)}},{\displaystyle \frac{2m}{n(mn+1)}}\right].$$

According to Corollary 1,

$$min\left({\displaystyle \frac{-m}{n}};{\displaystyle \frac{-2m}{n(mn+1)}};{\displaystyle \frac{-2m^2}{mn+1}}\right)⩾{\displaystyle \frac{-1}{\lambda }}.$$

Thus,

$$C_{\lambda }(u,v)=uv+\lambda (u-v)min(mu,{\displaystyle \frac{1-u}{n}})min(mv,{\displaystyle \frac{1-v}{n}}),$$

(13)

which defines a copula where $\lambda \in ]0,{\displaystyle \frac{1}{max(\frac{m}{n};\frac{2m}{n(mn+1)};\frac{2m^2}{mn+1})}}]$.

If we instead consider the framework in which m and n are negative, the bounds framing ${\mathcal{V}}_c$ remain unchanged.

Example 5.

Consider $g\left(t\right)=si{n}^n\left(\pi t\right),$ for all $t\in I$ where $n⩾1$;

• we have $g^{\prime }\left(t\right)=n\pi cos\left(\pi t\right)si{n}^{n-1}\left(\pi t\right)$. Then,

$$\begin{array}{ccc}\hfill {\mathcal{V}}_c& =& n\pi (si{n}^n\left(\pi u\right)cos\left(\pi v\right)si{n}^{n-1}\left(\pi v\right)-si{n}^n\left(\pi v\right)cos\left(\pi u\right)si{n}^{n-1}\left(\pi u\right))\hfill \\ & +& {\left(n\pi \right)}^2(u-v)cos\left(\pi u\right)si{n}^{n-1}\left(\pi u\right)cos\left(\pi v\right)si{n}^{n-1}\left(\pi v\right)\hfill \\ & =& si{n}^{n-1}\left(\pi u\right)si{n}^{n-1}\left(\pi v\right)[n\pi \left(sin\left(\pi u\right)cos\left(\pi v\right)-sin\left(\pi v\right)cos\left(\pi u\right)\right)\hfill \\ & +& {\left(n\pi \right)}^{2}cos\left(\pi u\right)cos\left(\pi v\right)]\hfill \\ & =& si{n}^{n-1}\left(\pi u\right)si{n}^{n-1}\left(\pi v\right)\left[n\pi sin\left(\pi (u-v)\right)+{\left(n\pi \right)}^{2}cos\left(\pi u\right)cos\left(\pi v\right)\right].\hfill \end{array}$$

The fact that

$$-1⩽u-v⩽1,\mathit{then}-1⩽sin\left(\pi \right(u-v\left)\right)⩽1,$$

and

$$0⩽sin\left(\pi x\right)⩽1,\mathit{for}\mathit{every}x\in [0,1],-1⩽cos\left(\pi x\right)⩽1,\mathit{for}\mathit{every}x\in [0,1],$$

leads to

$${\mathcal{V}}_c\in [-(n\pi +{\left(n\pi \right)}^2),(n\pi +{\left(n\pi \right)}^2)],$$

and, thus, $\mathit{for}\mathit{every}(u,v)\in I^2$, $n⩾1,$

$$C_{\lambda }(u,v)=uv+\lambda (u-v)si{n}^n\left(\pi u\right)si{n}^n\left(\pi v\right)$$

(14)

defines a copula, where $\lambda \in ]0,{\displaystyle \frac{1}{n\pi +{\left(n\pi \right)}^2}}]$.

3.3. An Asymmetric Extension of the FGM Family Through Parametric Perturbation

The Farlie–Gumbel–Morgenstern (FGM) copulas constitute a classical family in dependence modeling, appreciated for their mathematical simplicity and direct interpretability. However, their practical applicability is limited by a major constraint: their symmetric structure does not allow them to capture directional dependencies frequently observed in real data.

Our research addresses these limitations by proposing an innovative extension of the FGM family through a controlled parametric perturbation, in which the perturbation term simultaneously incorporates symmetric and asymmetric components.

We can easily see that our subclass defined by (12) serves to perturb any member of the semiparametric family of symmetric copulas introduced and studied by Rodrıguez-Lallena and Úbeda-Flores [8], and it is given by

$$C(u,v)=uv+\theta h\left(u\right)h\left(v\right),\mathrm{for}\mathrm{all}\theta \in [-1,1],$$

(15)

which includes the Farlie–Gumbel–Morgenstern family as a particular case.

Let $g\left(x\right)=x(1-x)$; then $g^{\prime }\left(x\right)=1-2x$ and, hence, $g\left(x\right)\in [0,g\left({\displaystyle \frac{1}{2}}\right)]=\left[0,{\displaystyle \frac{1}{4}}\right],\mathrm{for}\mathrm{every}x\in I$. Moreover, we have $g^{″}\left(x\right)=-2⩽0$, and therefore $g^{\prime }\left(x\right)\in [g^{\prime }\left(1\right),g^{\prime }\left(0\right)]=[-1,1],\mathrm{for}\mathrm{every}x\in I$. According to Corollary 1

$${\mathcal{V}}_c=u(1-u)(1-2v)-v(1-v)(1-2u)+(u-v)(1-2u)(1-2v)\in \left[-{\displaystyle \frac{3}{2}},{\displaystyle \frac{3}{2}}\right],$$

and it follows that

$$C(u,v)=uv\left(1+\lambda (u-v)(1-u)(1-v)\right),\mathrm{where}\lambda \in ]0,{\displaystyle \frac{2}{3}}].$$

(16)

A convex combination of the two parametric families of copulas (15) and (16) (the case where $h\left(x\right)=x(1-x)$) generates, for every $\alpha \in I$, the parametric family of copulas expressed by

$$\begin{array}{ccc}\hfill C_{\{\alpha ,\theta ,\lambda \}}(u,v)& =& uv+\alpha \theta uv(1-u)(1-v)+(1-\alpha )\lambda (u-v)uv(1-u)(1-v)\hfill \\ & =& uv\left(1+(\alpha \theta +(1-\alpha \left)\lambda \right(u-v\left)\right)(1-u)(1-v)\right),\hfill \end{array}$$

where $\theta \in [-1,1],\lambda \in ]0,{\displaystyle \frac{2}{3}}]$ and $\alpha \in I$.

Each copula of the FGM family can thus be perturbed by a convex combination with a copula that belongs to our parametric subclass, thus providing a richer modeling tool for capturing complex dependence structures observed in real data.

3.4. Illustrations

We conclude this section with a few plots to visualize the shapes of the two previous examples. More precisely, we provide the 3D surfaces and contour plots for the copulas previously defined in (13) and (14), as well as for their corresponding perturbation factors P.

As an illustration, we have chosen the parameters characterizing the copula defined in (13) as $m=1.5$ and $n=0.5$, whereas $n=1$ for the copula defined in (14) (see Figure 1). These visualizations make it possible to analyze how the perturbations modify the dependence structure between the variables, highlighting the deviations from the independence case in particular.

The 3D representations in Figure 1 display slightly curved hyperbolic surfaces compared to the independence copula $\Pi$, suggesting a weak dependence introduced by the two perturbations. Regarding the contour plots, their deviations from the copula $\Pi$ are subtle; they are not symmetric with respect to the diagonal $u=v$, reflecting the asymmetry of the two proposed examples.

Note that the larger the parameters m and n, the weaker the perturbation, bringing the perturbed copula closer to the independence copula.

4. Properties of the New Subclass

4.1. Preliminaries on Measures of Concordance

Let $(X_1,X_2)$ and $(Y_1,Y_2)$ be two continuous random vectors. If the corresponding copulas are $C_1$ and $C_2$, then the function Q is defined as follows:

$$Q(C_1,C_2)=4{\int }_{I^2}{C}_2(u,v)d{C}_1(u,v)-1,$$

(17)

which is called the concordance function, as introduced in [21]. Its probabilistic representation is given by

$$P((X_1-X_2)(Y_1-Y_2)>0)-P((X_1-X_2)(Y_1-Y_2)<0).$$

This function has several properties, among which we mention the following:

• Symmetry, $Q(C_1,C_2)=Q(C_2,C_1)$.
• Monotonicity in each argument (meaning that a stronger concordance ordering between copulas implies a higher concordance value).
  Formally, if $C_1⩽C_2$ and $C_2⩽C_3$, then

  $$Q(C_1,C_3)⩽Q(C_2,C_3)\mathrm{and}Q(C_1,C_2)⩽Q(C_1,C_3).$$
• The concordance function Q remains unchanged when the copulas are replaced by their survival copulas or by their transposes, while it changes sign when they are replaced by their reflections:

  $$Q(C_1,C_2)=Q(\varphi \left(C_1\right),\varphi \left(C_2\right))=Q(\pi \left(C_1\right),\pi \left(C_2\right)),$$

  $$Q({\varphi }_i\left(C_1\right),{\varphi }_i\left(C_2\right))=-Q(C_1,C_2)\mathrm{for}\mathrm{every}i\in \{1,2\}.$$
  For more details, see [18].

In [22], Scarsini offers a characterization of concordance measures $\kappa$ as mappings that assign a real number $\kappa \left(C\right)$ to each copula C, in accordance with axioms set forth in the following definition.

Definition 2.

A function $\kappa :C\to [-1,1]$ is called a concordance measure if it satisfies the following conditions:

(C₁)

$\kappa \left(\pi \right(C\left)\right)=\kappa \left(C\right)$;

(C₂)

$\kappa \left({\varphi }_1\left(C\right)\right)=\kappa \left({\varphi }_2\left(C\right)\right)=-\kappa \left(C\right)$;

(C₃)

$\kappa \left(M\right)=1$;

(C₄)

For every $C_1,C_2\in \mathcal{C}$ where $C_1⩽C_2$, $\kappa \left(C_1\right)⩽\kappa \left(C_2\right)$;

(C₅)

If a sequence of copulas ${\left(C_n\right)}_{n\in \mathbb{N}}$ converges uniformly to $C\in \mathcal{C}$, then $\underset{x\to +\infty }{lim}\kappa \left(C_n\right)=\kappa \left(C\right)$.

Since the independence copula $\Pi$ is invariant under both reflections, by condition $\left(C_2\right)$, we can deduce that

$$\kappa (\Pi )=0.$$

(18)

The four most frequently used concordance measures, which also play a central role, are Kendall’s $\tau$, Spearman’s $\rho$, Gini’s $\gamma$, and Blomqvist’s $\beta$.

When condition $\left(C2\right)$ is replaced with the normalization condition (18), we obtain what Liebscher [23] calls a weak concordance measure. Spearman’s footrule $\varphi$ is a typical example of such a weak measure of concordance (see [24]).

Note that the set of all weak concordance measures covers all concordance measures.

A (weak) concordance measure $C\to [-1,1]$ is said to be convex if it satisfies the following property:

$$\kappa (\theta C_1+(1-\theta )C_2)=\theta \kappa \left(C_1\right)+(1-\theta )\kappa \left(C_2\right)\mathrm{for}\mathrm{every}{C}_1,C_2\in \mathbf{C},\mathrm{for}\mathrm{all}\theta \in I,$$

which can be seen as the linearity of $\kappa$ applied to convex combinations of copulas. Let $l_{\kappa }:I\to [-1,1]$ be defined by $l_{\kappa }\left(\theta \right)=\kappa (\theta C_1+(1-\theta )C_2)$. $\kappa$ is a convex concordance measure if and only if $l_{\kappa }$ is a linear function; i.e., $l_{\kappa }\left(\theta \right)=\left(\kappa \left(C_1\right)-\kappa \left(C_2\right)\right)\theta +\kappa \left(C_2\right)$. Equivalently, it means that $\kappa$ is of degree 1 in accordance with Edwards and Taylor [25].

To simplify the statements, we will henceforth use the term concordance measure to refer to the five studied measures.

The first four concordance measures can be expressed in terms of the concordance function Q.

• Kendall’s $\tau$: $\tau \left(C\right)=Q(C,C)=12{\int }_{I^2}C(u,v)dC(u,v)-3$;
• Spearman’s $\rho$: $\rho \left(C\right)=3Q(C,\Pi )=4{\int }_{I^2}C(u,v)dudv-1$;
• Gini’s $\gamma$: $\gamma \left(C\right)=Q(C,M)+Q(C,W)=4{\int }_I(C(u,u)+C(u,1-u)du-2$;
• Spearman’s footrule $\varphi$: $\varphi \left(C\right)={\displaystyle \frac{1}{2}}(3Q(C,M)-1)=6{\int }_{I^2}C(u,u)du-2$;
• On the other hand, Blomqvist’s $\beta$ is given by

  $$\beta \left(C\right)=4C({\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}})-1.$$

Example 6.

For every $\theta \in [-1,1]$, the four dependence measures ρ, τ, β, and γ associated with FGM  copulas given by (2) are linear functions of the parameter θ.

$$\rho \left(C_{\theta }\right)={\displaystyle \frac{1}{3}}\theta \tau \left(C_{\theta }\right)={\displaystyle \frac{2}{9}}\theta \beta \left(C_{\theta }\right)={\displaystyle \frac{1}{4}}\theta \gamma \left(C_{\theta }\right)={\displaystyle \frac{4}{15}}\theta .$$

Furthermore, these parameters exhibit simple proportional relationships:

$${\displaystyle \frac{\rho \left(C_{\theta }\right)}{\tau \left(C_{\theta }\right)}}={\displaystyle \frac{3}{2}}{\displaystyle \frac{\rho \left(C_{\theta }\right)}{\beta \left(C_{\theta }\right)}}={\displaystyle \frac{4}{3}}{\displaystyle \frac{\rho \left(C_{\theta }\right)}{\gamma \left(C_{\theta }\right)}}={\displaystyle \frac{5}{4}},$$

and the ranges of these measures for any $\theta \in [-1,1]$ are given by

$$\rho \left(C_{\theta }\right)\in \left[-{\displaystyle \frac{1}{3}},{\displaystyle \frac{1}{3}}\right]\tau \left(C_{\theta }\right)\in \left[-{\displaystyle \frac{2}{9}},{\displaystyle \frac{2}{9}}\right]\beta \left(C_{\theta }\right)\in \left[-{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}}\right]\gamma \left(C_{\theta }\right)\in \left[-{\displaystyle \frac{4}{15}},{\displaystyle \frac{4}{15}}\right].$$

4.2. Single Point-Generated Convex Weak Concordance Measures

We now turn our attention to a concordance measure, $\mathsf{s}$, whose values depend on the symmetrized copula $C_s$ given previously by (5). This measure encompasses several well-known (weak) convex concordance measures from the literature, including Spearman’s Rho, Gini’s Gamma, Blomqvist’s Beta, and Spearman’s footrule.

Since all copulas coincide on the boundary of the unit square, we consider a point $(u,v)\in {]0,1[}^2$. The fact that $C_s$ is a symmetric copula implies that $C_s(u,v)=C_s(v,u),C_s(1-u,v)=C_s(v,1-u),C_s(u,1-v)=C_s(1-v,u),$ and $C_s(1-u,1-v)=C_s(1-v,1-u)$, which allows us to focus only on points within the set $\Delta =\left\{\right(u,v)\in I|v⩽u\}$. Given any fixed point $(u,v)\in \Delta$, the mapping $C_s:\mathcal{C}\to \mathbb{R}$ is increasing—i.e., for every $C_1⩽C_2$, we have $C_s\left(C_1\right)⩽C_s\left(C_2\right)$—and, consequently, $C_s(\Pi )=uv⩽v=C_s\left(M\right)$. We can therefore normalize $C_s$ into $\mathsf{s}$ as follows:

$${\mathsf{s}}_{(u,v)}={\displaystyle \frac{{C_s}_{(u,v)}\left(C\right)-{C_s}_{(u,v)}(\Pi )}{{C_s}_{(u,v)}\left(M\right)-{C_s}_{(u,v)}(\Pi )}}.$$

Based on the above considerations, we introduce an alternative formulation for the family of convex concordance measures generated by a single point, recently proposed in [14].

Theorem 3.

Let $(u,v)\in \Delta$. The function ${\mathsf{s}}_{(u,v)}:\mathcal{C}\to \mathbb{R}$ given by

$${\mathsf{s}}_{(u,v)}\left(C\right)={\displaystyle \frac{{C_s}_{(u,v)}\left(C\right)-uv}{v-uv}},$$

(19)

is a weak convex concordance measure.

Proof. 

By construction, we immediately observe that ${\mathsf{s}}_{(u,v)}\left(M\right)=1$ and ${\mathsf{s}}_{(u,v)}(\Pi )=0$. Moreover, for all ${\left(C_n\right)}_{n⩾1}\subset \mathcal{C}$, if $C_n\to \mathcal{C}$, then ${\mathsf{s}}_{(u,v)}\left(C_n\right)\to {\mathsf{s}}_{(u,v)}\left(C\right)$ uniformly for each $(u,v)\in \Delta$. From our earlier discussion, the other axioms of weak concordance measures (invariance under permutation and boundedness on $I^2$) are easily verified.

Regarding the convexity of $\mathsf{s}$ we have, for every $C_1,C_2\in \mathcal{C}$ and for all $\lambda \in I$,

$$\begin{array}{ccc}& & {C_s}_{(u,v)}(\lambda C_1+(1-\lambda )C_2)\hfill \\ & =& {\displaystyle \frac{C_s(\lambda C_1+(1-\lambda )C_2)(u,v)-uv}{v-uv}}\hfill \\ & =& {\displaystyle \frac{\lambda (C_1(u,v)+\pi \left(C_1\right)(u,v))+(1-\lambda )(C_2(u,v)+\pi \left(C_2\right)(u,v))}{v-uv}}\hfill \\ & =& \lambda {\displaystyle \frac{C_1(u,v)+\pi \left(C_1\right)(u,v)-uv}{v-uv}}+(1-\lambda ){\displaystyle \frac{C_2(u,v)+\pi \left(C_2\right)(u,v)-uv}{v-uv}}\hfill \\ & =& \lambda {C_s}_{(u,v)}\left(C_1\right)+(1-\lambda ){C_s}_{(u,v)}\left(C_2\right).\hfill \end{array}$$

Thus, $\mathsf{s}$ is convex. We omit the detailed calculations. □

Remark 3.

For $C=W,$ for every $(u,v)\in \Delta$

$${\mathsf{s}}_{(u,v)}\left(W\right)=\left\{\begin{array}{ccc}{\displaystyle \frac{-u}{1-u}},\hfill & if& u+v⩽1;\hfill \\ {\displaystyle \frac{v-1}{v}},\hfill & if& u+v⩾1.\hfill \end{array}\right.$$

which yields ${\mathsf{s}}_{(u,v)}\left(W\right)\in ]-\infty ,0[$,

• which confirms that there is no lower bound for convex weak concordance measures.

Example 7.

For any fixed point $(u,v)\in S=\left\{\right(a,b)\in \Delta |a=b\}$, we have

$${\mathsf{s}}_{(u,u)}\left(C\right)={\displaystyle \frac{{C_s}_{(u,u)}\left(C\right)-u^2}{u-u^2}}.$$

When $u={\displaystyle \frac{1}{2}}$, we obtain Blomqvist’s $\beta$; i.e., ${\mathsf{s}}_{({\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}})}\left(C\right)=\beta$.

4.3. Representation of Basic Weak Convex Concordance Measures via $s^P$ and $s_{\Psi }$

Any convex combination of (weak convex) concordance measures remains a (weak convex) concordance measure.

Formally, let ${\mathsf{s}}_{(u_1,v_1)},{\mathsf{s}}_{(u_2,v_2)},\cdots ,{\mathsf{s}}_{(u_k,v_k)}$ be weak convex concordance measures. Then, the function $\mathsf{s}:\mathcal{C}\to \mathbb{R}$ defined by $\mathsf{s}\left(C\right)={\sum }_{i=1}^k{\lambda }_i{\mathsf{s}}_{(u_i,v_i)}\left(C\right)$, where ${\lambda }_i\in I$ with ${\sum }_{i=1}^k{\lambda }_i=1$, is also a concordance measure. This can be interpreted as a Lebesgue integral with respect to a probability measure $\Psi$ defined on $\mathcal{B}(\Delta )$ (the Borel subsets of I) by $\Psi (\Omega )={\sum }_{(u_i,v_i)\in \Omega }{\lambda }_i$; i.e.,

$${\mathsf{s}}_{\Psi }\left(C\right)={\int }_{\Delta }{\mathsf{s}}_{(u,v)}\left(C\right)d\Psi (u,v).$$

(20)

Theorem 4.

Let P be a probability measure defined on $(\Delta ,\mathcal{B}(\Delta \left)\right)$. Then, the mapping $s^P$ given by

$${\mathsf{s}}^P\left(C\right)={\displaystyle \frac{{\int }_{\Delta }{C_s}_{(u,v)}\left(C\right)dP(u,v)-{\int }_{\Delta }uvdP(u,v)}{{\int }_{\Delta }vdP(u,v)-{\int }_{\Delta }uvdP(u,v)}}$$

(21)

is a convex weak concordance measure.

Proof. 

The proof of this statement is similar to that of Theorem 3. □

Moreover, note that there exists a probability measure $\Psi$ on $(\Delta ,\mathcal{B}(\Delta \left)\right)$ such that, for each copula $C\in \mathcal{C}$, we have ${\mathsf{s}}_{\Psi }\left(C\right)={\mathsf{s}}^P\left(C\right)$, where ${\mathsf{s}}_{\Psi }$ and ${\mathsf{s}}^P$ are the weak convex concordance measures given in (20) and (21), respectively (see [14], Theorem 4.1).

Proposition 3.

Let P and Ψ be two probability measures on $(\Delta ,\mathcal{B}(\Delta \left)\right)$, with respective densities p and ψ.

• If $p(u,v)=2$ and $\psi (u,v)=24v(1-u)$, $\mathit{for}\mathit{all}(u,v)\in \Delta$,
  then ${\mathsf{s}}^P={\mathsf{s}}_{\Psi }=\rho$ (Spearman’s ρ).
• If $p(u,v)=\left\{\begin{array}{ccc}{\displaystyle \frac{1}{2}},\hfill & if& (u,v)\in S_1;\hfill \\ 1,\hfill & if& (u,v)\in S_2,\hfill \end{array}\right.$
  and $\psi (u,v)=\left\{\begin{array}{ccc}4v(1-u),\hfill & if& (u,v)\in S_1;\hfill \\ 8v(1-u),\hfill & if& (u,v)\in S_2,\hfill \end{array}\right.$
  where $S_1=\{v=u,u\in ]0,1\left[\right\}$ and $S_2=\{v=1-u,u\in ]{\displaystyle \frac{1}{2}},1\left[\right\}$,
  then ${\mathsf{s}}^P={\mathsf{s}}_{\Psi }=\gamma$ (Gini’s γ).
• If $p(u,v)=\left\{\begin{array}{ccc}1,\hfill & if& (u,v)\in S_1;\hfill \\ 0,\hfill & & otherwise,\hfill \end{array}\right.$
  and $\psi (u,v)=\left\{\begin{array}{ccc}6u(1-u),\hfill & if& (u,v)\in S_1;\hfill \\ 0,\hfill & & otherwise,\hfill \end{array}\right.$
  then ${\mathsf{s}}^P={\mathsf{s}}_{\Psi }=\varphi$ (Spearman’s footrule ϕ).

Proof. 

• Consider the probability measure $\Psi$ on $(\Delta ,\mathcal{B}(\Delta \left)\right)$ with density $\psi (u,v)=24v(1-u),(u,v)\in \Delta$, as $d\Psi (u,v)=24v(1-u)dudv$. Then, the corresponding convex weak concordance measure ${\mathsf{s}}_{\Psi }$ introduced in (20) can be determined as follows:

  $$\begin{array}{ccc}\hfill {\mathsf{s}}_{\Psi }\left(C\right)& =& {\int }_{\Delta }{\displaystyle \frac{{C_s}_{(u,v)}\left(C\right)-uv}{v-uv}}d\Psi (u,v)\hfill \\ & =& 24{\int }_{\Delta }{C_s}_{(u,v)}\left(C\right)dudv-24{\int }_0^1{\int }_0^{u}uvdudv\hfill \\ & =& 12{\int }_{\Delta }C(u,v)+\pi \left(C\right)(u,v)dudv-12{\int }_0^1{u}^{3}du\hfill \\ & =& 12{\int }_{I^2}C(u,v)dudv-3\hfill \\ & =& \rho \left(C\right).\hfill \end{array}$$
  The same result can be obtained using the representation $s^P$ given in (21).
• Let $\Psi$ be a probability measure on $(\Delta ,\mathcal{B}(\Delta \left)\right)$ whose support is the set $S=\{v=u,u\in ]0,1\left[\right\}\bigcup \{v=1-u,u\in ]{\displaystyle \frac{1}{2}},1\left[\right\}$ and density $\psi (u,v)=\left\{\begin{array}{ccc}4v(1-u),\hfill & if& (u,v)\in S_1;\hfill \\ 8v(1-u),\hfill & if& (u,v)\in S_2.\hfill \end{array}\right.$
  Then,

  $$\begin{array}{ccc}\hfill {\mathsf{s}}_{\Psi }\left(C\right)& =& 4{\int }_0^1{C_s}_{(u,u)}\left(C\right)-u^{2}du+8{\int }_{1/2}^1{C_s}_{(u,1-u)}\left(C\right)-u(1-u)du\hfill \\ & =& 4{\int }_0^{1}C(u,u)du+4{\int }_{1/2}^{1}C(u,1-u)+C(1-u,u)du\hfill \\ & & -4{\int }_0^1{u}^{2}du-8{\int }_{1/2}^{1}u(1-u)du\hfill \\ & =& 4{\int }_0^{1}C(u,u)du+4{\int }_0^{1}C(u,1-u)du-{\displaystyle \frac{4}{3}}-{\displaystyle \frac{2}{3}}\hfill \\ & =& 4{\int }_0^{1}C(u,u)+C(u,1-u)du-2\hfill \\ & =& \gamma \left(C\right).\hfill \end{array}$$
• The support of the probability measure $\Psi$ is the set $S=\{v=u,u\in ]0,1\left[\right\}$. We directly obtain

  $$\begin{array}{ccc}\hfill {\mathsf{s}}_{\Psi }\left(C\right)& =& 6{\int }_0^1{C_s}_{(u,u)}\left(C\right)-u^{2}du\hfill \\ & =& 6{\int }_0^{1}C(u,u)du-6{\int }_0^1{u}^{2}du\hfill \\ & =& 6{\int }_0^{1}C(u,u)du-2\hfill \\ & =& \varphi \left(C\right). \square \hfill \end{array}$$

4.4. Dependence Properties

4.4.1. Measures of Concordance

In this section, we focus on the evaluation and analysis of the behavior of several standard measures of concordance for the parametric subclass of copula families $C_{\theta }$ defined by

$$C_{\theta }(u,v)=uv+\theta (u-v)g\left(u\right)g\left(v\right),$$

assuming that the necessary conditions on g are satisfied and that $\theta$ is a real parameter depending on the choice of g, whose set of admissible values is restricted by the 2-increasing condition of $C_{\theta }$.

We adopt an inductive approach by considering the weak convex measure of concordance ${\mathsf{s}}_{\Psi }$ given by (20), which encompasses the usual measures of concordance ($\rho ,\beta ,\gamma$, and $\varphi$). This unified method makes it possible to show easily that, for any weak convex measure of concordance, its value is necessarily zero on our subclass, whereas Kendall’s $\tau$ may take non-zero values. Its expression is established in the proposition below.

This distinctive property illustrates the richness of the subclass and justifies its relevance for modeling complex dependence structures.

Let us recall that by applying integration by parts to the expression of Kendall’s $\tau$ defined by

$${\tau }_C=4{\int }_{I^2}C(u,v)dC(u,v)-1,$$

we obtain an alternative formulation that is more convenient for numerical computation:

$${\tau }_C=1-4{\int }_{I^2}{\displaystyle \frac{\partial C(u,v)}{\partial u}}{\displaystyle \frac{\partial C(u,v)}{\partial v}}dudv.$$

Furthermore, if the copula C is absolutely continuous, then

$$dC(u,v)={\displaystyle \frac{{\partial }^{2}C(u,v)}{\partial u\partial v}}dudv=c(u,v)dudv,$$

where $c(u,v)$ denotes the copula density of C. For more detail, we refer the reader to [16].

Proposition 4.

Let $(X,Y)$ be a random pair whose copula $C_{\theta }$ is given by (12). We have

$$\rho \left(C_{\theta }\right)=\gamma \left(C_{\theta }\right)=\beta \left(C_{\theta }\right)=\varphi \left(C_{\theta }\right)=0.$$

Kendall’s τ admits the following representation:

$${\tau }_{\theta }=2{\theta }^2{\left({\int }_I{g}^2\left(u\right)du\right)}^2.$$

Proof. 

For every copula $C_{\theta }$ belonging to our new subclass of copulas of the form (12), we trivially observe that

$${\mathsf{s}}_{(u,v)}\left(C_{\theta }\right)={\displaystyle \frac{{C_s}_{(u,v)}\left(C_{\theta }\right)-uv}{v-uv}}=0.$$

This result follows directly from the characterizing property (6) of this subclass, which makes it possible to obtain

$${\mathsf{s}}_{\Psi }\left(C_{\theta }\right)={\int }_{\Delta }{\mathsf{s}}_{(u,v)}\left(C_{\theta }\right)d\Psi (u,v)=0.$$

Consequently, for appropriate probability measures (see Proposition 3), the classical measures of concordance—($\rho ,\gamma ,\beta$, and $\varphi$)—vanish identically.

Regarding Kendall’s $\tau$, the partial derivatives of the copula $C_{\theta }$ are expressed by

$$\left\{\begin{array}{ccc}{\displaystyle \frac{\partial C_{\theta }(u,v)}{\partial u}}\hfill & =& v+\theta (g\left(u\right)g\left(v\right)+(u-v)g^{\prime }\left(u\right)g\left(v\right));\hfill \\ {\displaystyle \frac{\partial C_{\theta }(u,v)}{\partial v}}\hfill & =& u+\theta (-g\left(u\right)g\left(v\right)+(u-v)g\left(u\right)g^{\prime }\left(v\right)).\hfill \end{array}\right.$$

Kendall’s $\tau$ can then be written as follows:

$${\tau }_{\theta }=1-4\mathfrak{I},\mathrm{where}\mathfrak{I}={\int }_{I^2}{\displaystyle \frac{\partial C_{\theta }(u,v)}{\partial u}}{\displaystyle \frac{\partial C_{\theta }(u,v)}{\partial v}}dudv,$$

$$\begin{array}{ccc}\hfill \mathfrak{I}& =& {\int }_{I^2}(v+\theta (g\left(u\right)g\left(v\right)+(u-v)g^{\prime }\left(u\right)g\left(v\right))(u+\theta (-g\left(u\right)g\left(v\right)+(u-v)g\left(u\right)g^{\prime }\left(v\right))dudv\hfill \\ & =& {\int }_{I^2}uvdudv-\theta {\int }_{I^2}vg\left(v\right)g\left(u\right)dudv+\theta {\int }_{I^2}v(u-v)g\left(u\right)g^{\prime }\left(v\right)dudv\hfill \\ & +& \theta {\int }_{I^2}ug\left(u\right)g\left(v\right)dudv-{\theta }^2{\int }_{I^2}{g}^2\left(u\right)g^2\left(v\right)dudv+{\theta }^2{\int }_{I^2}(u-v)g{\left(u\right)}^2{g}^{\prime }\left(v\right)g\left(v\right)dudv\hfill \\ & +& \theta {\int }_{I^2}u(u-v)g\left(v\right)g^{\prime }\left(u\right)dudv-{\theta }^2{\int }_{I^2}(u-v)g^{\prime }\left(u\right)g\left(u\right)g^2\left(v\right)dudv\hfill \\ & +& {\theta }^2{\int }_{I^2}{(u-v)}^2{g}^{\prime }\left(u\right)g\left(u\right)g^{\prime }\left(v\right)g\left(v\right)dudv,\hfill \end{array}$$

given that

$${\int }_{I^2}vg\left(v\right)g\left(u\right)dudv={\int }_{I}g\left(u\right)du{\int }_{I}vg\left(v\right)dv={\int }_{I}g\left(v\right)dv{\int }_{I}ug\left(u\right)du={\int }_{I^2}ug\left(u\right)g\left(v\right)dudv,$$

Hence,

$${\int }_{I^2}vg\left(v\right)g\left(u\right)dudv={\int }_{I^2}ug\left(u\right)g\left(v\right)dudv.$$

(22)

Analogously, we find that

$$\begin{array}{ccc}\hfill {\int }_{I^2}v(u-v)g\left(u\right)g^{\prime }\left(v\right)dudv& =& {\int }_{I^2}uvg\left(u\right)g^{\prime }\left(v\right)dudv-{\int }_{I^2}{v}^{2}g\left(u\right)g^{\prime }\left(v\right)dudv\hfill \\ & =& {\int }_{I}ug\left(u\right)du{\int }_{I}v{g}^{\prime }\left(v\right)dv-{\int }_{I}g\left(u\right)du{\int }_I{v}^2{g}^{\prime }\left(v\right)dv\hfill \\ & =& {\int }_{I}vg\left(v\right)dv{\int }_{I}u{g}^{\prime }\left(u\right)du-{\int }_{I}g\left(v\right)dv{\int }_I{u}^2{g}^{\prime }\left(u\right)du\hfill \\ & =& {\int }_{I^2}uvg\left(v\right)g^{\prime }\left(u\right)dudv-{\int }_{I^2}{u}^{2}g\left(v\right)g^{\prime }\left(u\right)dudv\hfill \\ & =& {\int }_{I^2}u(v-u)g\left(v\right)g^{\prime }\left(u\right)dudv,\hfill \end{array}$$

Hence,

$${\int }_{I^2}v(u-v)g\left(u\right)g^{\prime }\left(v\right)dudv=-{\int }_{I^2}u(u-v)g\left(v\right)g^{\prime }\left(u\right)dudv.$$

(23)

The two equalities (22) and (23) therefore reveal that four terms of $\mathfrak{I}$ are pairwise opposite, hence their mutual cancelation.

Moreover,

$${\int }_{I^2}uvdudv={\int }_I{\displaystyle \frac{u}{2}}du={\displaystyle \frac{1}{4}},\mathrm{then}1-4{\int }_{I^2}uvdudv=0.$$

It follows that Kendall’s $\tau$ simplifies and can be expressed as a sum of four terms, an expression that will be further simplified subsequently.

$${\tau }_{\theta }=4{\theta }^2({\mathfrak{I}}_1-{\mathfrak{I}}_2+{\mathfrak{I}}_3-{\mathfrak{I}}_4),$$

(24)

where

$$\begin{array}{ccc}\hfill {\mathfrak{I}}_1& =& {\int }_{I^2}{g}^2\left(u\right)g^2\left(v\right)dudv,\hfill \\ \hfill {\mathfrak{I}}_2& =& {\int }_{I^2}(u-v)g{\left(u\right)}^2{g}^{\prime }\left(v\right)g\left(v\right)dudv,\hfill \\ \hfill {\mathfrak{I}}_3& =& {\int }_{I^2}(u-v)g^{\prime }\left(u\right)g\left(u\right)g^2\left(v\right)dudv,\hfill \\ \hfill {\mathfrak{I}}_4& =& {\int }_{I^2}{(u-v)}^2{g}^{\prime }\left(u\right)g\left(u\right)g^{\prime }\left(v\right)g\left(v\right)dudv.\hfill \end{array}$$

On one hand, we have

$$\begin{array}{ccc}\hfill {\mathfrak{I}}_2& =& {\int }_{I^2}(u-v)g{\left(u\right)}^2{g}^{\prime }\left(v\right)g\left(v\right)dudv\hfill \\ & =& {\int }_{I^2}ug{\left(u\right)}^2{g}^{\prime }\left(v\right)g\left(v\right)dudv-{\int }_{I^2}vg{\left(u\right)}^2{g}^{\prime }\left(v\right)g\left(v\right)dudv\hfill \\ & =& {\int }_{I}ug{\left(u\right)}^{2}du{\int }_I{g}^{\prime }\left(v\right)g\left(v\right)dv-{\int }_{I}g{\left(u\right)}^{2}du{\int }_{I}v{g}^{\prime }\left(v\right)g\left(v\right)dv\hfill \\ & =& {\int }_{I}vg{\left(v\right)}^{2}dv{\int }_I{g}^{\prime }\left(u\right)g\left(u\right)du-{\int }_{I}g{\left(v\right)}^{2}dv{\int }_{I}u{g}^{\prime }\left(u\right)g\left(u\right)du\hfill \\ & =& {\int }_{I^2}vg{\left(v\right)}^2{g}^{\prime }\left(u\right)g\left(u\right)dudv-{\int }_{I^2}ug{\left(v\right)}^2{g}^{\prime }\left(u\right)g\left(u\right)dudv\hfill \\ & =& -{\int }_{I^2}(u-v)g{\left(v\right)}^2{g}^{\prime }\left(u\right)g\left(u\right)dudv\hfill \end{array}$$

and, consequently,

$$\begin{array}{ccc}\hfill {\mathfrak{I}}_3-{\mathfrak{I}}_2& =& {\int }_{I^2}(u-v)g^{\prime }\left(u\right)g\left(u\right)g^2\left(v\right)dudv-{\int }_{I^2}(u-v)g{\left(u\right)}^2{g}^{\prime }\left(v\right)g\left(v\right)dudv\hfill \\ & =& {\int }_{I^2}(u-v)g^{\prime }\left(u\right)g\left(u\right)g^2\left(v\right)dudv+{\int }_{I^2}(u-v)g{\left(v\right)}^2{g}^{\prime }\left(u\right)g\left(u\right)dudv\hfill \\ & =& 2{\int }_{I^2}(u-v)g^{\prime }\left(u\right)g\left(u\right)g^2\left(v\right)dudv\hfill \\ & =& 2{\int }_{I^2}u{g}^{\prime }\left(u\right)g\left(u\right)g^2\left(v\right)dudv-2{\int }_{I^2}vg{\left(v\right)}^2{g}^{\prime }\left(u\right)g\left(u\right)dudv.\hfill \end{array}$$

On the other hand,

$${\int }_{I^2}{g}^2\left(u\right)g^2\left(v\right)dudv={\int }_I{g}^2\left(v\right)\left({\int }_I{g}^2\left(u\right)du\right)dv,$$

since g vanishes at the boundaries ($g\left(0\right)=g\left(1\right)=0$), and an integration by parts yields

$${\int }_I{g}^2\left(u\right)du={\left[u{g}^2\left(u\right)\right]}_0^1-2{\int }_{I}u{g}^{\prime }\left(u\right)g\left(u\right)du=-2{\int }_{I}u{g}^{\prime }\left(u\right)g\left(u\right)du,$$

Thus,

$${\mathfrak{I}}_1={\int }_{I^2}{g}^2\left(u\right)g^2\left(v\right)dudv=-2{\int }_{I^2}u{g}^{\prime }\left(u\right)g\left(u\right)g^2\left(v\right)dudv,$$

Therefore,

$$\begin{array}{ccc}\hfill {\mathfrak{I}}_1+{\mathfrak{I}}_3-{\mathfrak{I}}_2& =& -2{\int }_{I^2}vg{\left(v\right)}^2{g}^{\prime }\left(u\right)g\left(u\right)dudv.\hfill \end{array}$$

(25)

Moreover, the last term of $\mathfrak{I}$ can be decomposed as follows:

$$\begin{array}{ccc}\hfill {\mathfrak{I}}_4& =& {\int }_{I^2}{(u-v)}^2{g}^{\prime }\left(u\right)g\left(u\right)g^{\prime }\left(v\right)g\left(v\right)dudv\hfill \\ & =& {\int }_{I^2}{u}^2{g}^{\prime }\left(u\right)g\left(u\right)g^{\prime }\left(v\right)g\left(v\right)dudv-2{\int }_{I^2}uv{g}^{\prime }\left(u\right)g\left(u\right)g^{\prime }\left(v\right)g\left(v\right)dudv\hfill \\ & +& {\int }_{I^2}{v}^2{g}^{\prime }\left(v\right)g\left(v\right)g^{\prime }\left(u\right)g\left(u\right)dudv,\hfill \end{array}$$

since

$${\int }_{I^2}{u}^2{g}^{\prime }\left(u\right)g\left(u\right)g^{\prime }\left(v\right)g\left(v\right)dudv={\int }_{I^2}{v}^2{g}^{\prime }\left(v\right)g\left(v\right)g^{\prime }\left(u\right)g\left(u\right)dudv,$$

and

$${\int }_{I^2}uv{g}^{\prime }\left(u\right)g\left(u\right)g^{\prime }\left(v\right)g\left(v\right)dudv={\int }_{I}u{g}^{\prime }\left(u\right)g\left(u\right)du{\int }_{I}v{g}^{\prime }\left(v\right)g\left(v\right)dv={\left({\int }_{I}u{g}^{\prime }\left(u\right)g\left(u\right)du\right)}^2,$$

Hence,

$$\begin{array}{ccc}\hfill {\mathfrak{I}}_4& =& 2{\int }_{I^2}{u}^2{g}^{\prime }\left(u\right)g\left(u\right)g^{\prime }\left(v\right)g\left(v\right)dudv-2{\int }_{I^2}uv{g}^{\prime }\left(u\right)g\left(u\right)g^{\prime }\left(v\right)g\left(v\right)dudv,\hfill \\ & =& {\int }_I{g}^{\prime }\left(v\right)g\left(v\right)dv{\int }_{I}2u^2{g}^{\prime }\left(u\right)g\left(u\right)du-2{\left({\int }_{I}u{g}^{\prime }\left(u\right)g\left(u\right)du\right)}^2,\hfill \end{array}$$

using integration by parts, and as $g\left(0\right)=g\left(1\right)=0$, we have

$$\begin{array}{ccc}\hfill {\mathfrak{I}}_4& =& {\int }_I{g}^{\prime }\left(v\right)g\left(v\right)\left({\left[u^{2}g{\left(u\right)}^2\right]}_0^1-2{\int }_{I}ug{\left(u\right)}^{2}du\right)dv-2{\left({\left[{\displaystyle \frac{1}{2}}ug{\left(u\right)}^2\right]}_0^1-{\displaystyle \frac{1}{2}}{\int }_I{g}^2\left(u\right)du\right)}^2\hfill \\ & =& -2{\int }_{I^2}ug{\left(u\right)}^2{g}^{\prime }\left(v\right)g\left(v\right)dudv-{\displaystyle \frac{1}{2}}{\left({\int }_I{g}^2\left(u\right)du\right)}^2.\hfill \end{array}$$

(26)

Combining (25) and (26) into (24), we obtain

$${\tau }_{\theta }=2{\theta }^2{\left({\int }_I{g}^2\left(u\right)du\right)}^2. \square$$

It is not surprising that Kendall’s tau behaves differently from the other concordance measures $\rho ,\gamma ,\beta$, and $\varphi$. This dissimilarity is explained by the linearity of the latter under convex combinations, whereas Kendall’s $\tau$ exhibits quadratic behavior.

Example 8.

We revisit the copula generated by the function $g_{\lambda }\left(x\right)={\lambda }^{1/2}sin{\left(\pi x\right)}^n$, where $\lambda \in ]0,{\displaystyle \frac{1}{n\pi +{\left(n\pi \right)}^2}}]$, as given by (14), for different values of the shape parameter n, which controls the concentration of mass around $x=0.5$, as shown in Figure 2.

• Figure 2 also illustrates the evolution of ${{\tau }_C}_{\theta }$, whose graph displays three curves, each corresponding to a specific value of n and its associated ${\theta }_{max}$. The vertical dashed lines indicate the value of ${\theta }_{max}$ for each n.

• For $n=1$, ${\theta }_{max}$ is the largest (approximately 0.0769).
• For the other two cases, $n=1.5$ and $n=2$, ${\theta }_{max}$ decreases (approximately 0.0371 and 0.0219, respectively).

Consequently, the larger the value of n, the smaller the value of θ and the lower the maximum value of ${\tau }_{C_{\theta }}$. These observations are consistent with the behavior of the function g: as n increases, g becomes more concentrated around $u=0.5$ and vanishes more rapidly elsewhere, which leads to a reduction in $I\left(\theta \right)$.

4.4.2. Positive Dependence

We now study a partial order on $C\in \mathcal{C}$. Let $C_1$ and $C_2$ be two copulas. $C_2$ is said to be more concordant than $C_1$, denoted by $C_1⪯C_2$, if $C_1(u,v)⩽C_2(u,v),$ for all $(u,v)\in I^2$ (see [16]). Let $C_1$ and $C_2$ be two copulas such that $C_1(u,v)=uv+{\theta }_1(u-v)g_1\left(u\right)g_2\left(v\right)$ and $C_2(u,v)=uv+{\theta }_2(u-v)f_1\left(u\right)f_2\left(v\right),$ for every $(u,v)\in I^2$, where ${\theta }_1>0$ and ${\theta }_2>0$. Then, $C_1⪯C_2$ if and only if

$$\left\{\begin{array}{ccc}{g}_1\left(u\right)g_2\left(v\right)⩽f_1\left(u\right)f_2\left(v\right),\hfill & si& u⩾v;\hfill \\ g_1\left(u\right)g_2\left(v\right)⩾f_1\left(u\right)f_2\left(v\right),\hfill & si& u⩽v.\hfill \end{array}\right.$$

The following two theorems consider continuous random pairs $(X,Y)$ associated with the copula $C\in \mathcal{C}$ and characterize the pairs that satisfy certain well-known positive dependence properties.

In what follows, we will use some concepts of positive dependence (for more details, see [16]). A copula C is said to be positively quadrant-dependent ($\mathbf{PQD}$) if $C⩾\Pi$ and negatively quadrant-dependent ($\mathbf{NQD}$) if $C⩽\Pi$. It should be noted that copulas exhibiting this property concentrate more mass near the main diagonal than near the opposite diagonal.

For our own parametric family of copulas given by (12) we have the following result, the proof of which is straightforward.

Theorem 5.

Let $\theta >0$ and $(X,Y)$ be a continuous random pair associated with the copula $C_{\theta }$ given by (12). Then, X and Y are PQD if and only if either g(u) and g(v) have the same sign for every $(u,v)\in {\Delta }_1=\{(x,y)\in I^2|x⩾y\}$ or g(u) and g(v) have opposite signs for every $(u,v)\in {\Delta }_2=\{(x,y)\in I^2|x⩽y\}$.

The interconnections between the dependence properties described in the following and a probabilistic interpretation of these relationships are discussed in [26]. For this, each concept of positive dependence studied in the following theorem implies PQD. We may therefore assume, without loss of generality, that the generating function g is concave with $g\left(x\right)=g(1-x),$ for all $x\in I$, which allows us to deduce that

$$g\left(x\right)⩾0,\mathrm{for}\mathrm{all}x\in I,$$

(27)

and that g is symmetric with respect to $v={\displaystyle \frac{1}{2}}$. Consequently, $g^{\prime }\left(x\right)⩾0,$ for all $x\in \left[0,{\displaystyle \frac{1}{2}}\right]$, and $g^{\prime }\left(x\right)⩽0,$ for all $x\in \left[{\displaystyle \frac{1}{2}},1\right]$. Under these considerations, we find that $C_{\theta }$ is PQD for every $(u,v)\in {\Delta }_1$.

This result is perfectly illustrated by Figure 1b,d, which provide a particularly insightful graphical representation.

Theorem 6.

Let $\theta >0$ and $(X,Y)$ be a continuous random pair associated with the copula $C_{\theta }$ given by (12), where g is a concave function with $g\left(x\right)=g(1-x)$ for any $x\in I$. Then:

1. 

Y is left-tail decreasing in X ($\mathit{LTD}\mathit{\left(}\mathit{Y}\right|\mathit{X})$) if and only if $u{g}^{\prime }\left(u\right)⩽g\left(u\right)⩽u(1-u)g^{\prime }\left(u\right)$ for almost all u, and $\mathit{LTD}\left(\mathit{X}\right|\mathit{Y})$ holds if and only if either $g^{\prime }\left(v\right)⩾0$ and $g\left(v\right)⩾v{g}^{\prime }\left(v\right)$ or $v(1-v)g^{\prime }\left(v\right)⩽g\left(v\right)⩽v{g}^{\prime }\left(v\right)$ for almost all v.

2. 

Y is right-tail increasing in X ($\mathit{RTI}\left(\mathit{Y}\right|\mathit{X})$) if and only if either $g^{\prime }\left(u\right)⩽0$ and $g\left(u\right)⩾(u-1)g^{\prime }\left(u\right)$ or $u(1-u)g^{\prime }\left(u\right)⩽g\left(u\right)⩽(u-1)g^{\prime }\left(u\right)$ for almost all u, and ($\mathit{RTI}\left(\mathit{X}\right|\mathit{Y})$) holds if and only if $(v-1)g^{\prime }\left(v\right)⩽g\left(v\right)⩽v(v-1)g^{\prime }\left(v\right)$ for almost all v.

3. 

Y is stochastically increasing in X ($\mathit{SI}\left(\mathit{Y}\right|\mathit{X})$) if and only if, for any $v\in I$, $g^{\prime }\left(u\right)⩽{\displaystyle \frac{1-u}{2}}{g}^{″}\left(u\right)$ for almost all u, and ($\mathit{SI}\left(\mathit{X}\right|\mathit{Y})$) holds if and only if, for any $u\in I$, $g^{\prime }\left(v\right)⩾{\displaystyle \frac{-v}{2}}{g}^{″}\left(v\right)$ for almost all v.

Proof. 

• According to (Corollary 5.2.6, [16]), $\mathbf{LTD}\left(\mathbf{Y}\right|\mathbf{X})$ holds if and only if, for any $v\in I$,

  $${\displaystyle \frac{\partial C(u,v)}{\partial u}}⩽{\displaystyle \frac{C(u,v)}{u}},\mathrm{for}\mathrm{almost}\mathrm{all}u.$$

  (28)
  Thus, we have ${\displaystyle \frac{\partial C_{\theta }(u,v)}{\partial u}}=v+\theta (g\left(u\right)g\left(v\right)+(u-v)g^{\prime }\left(u\right)g\left(v\right))$.
  In addition, ${\displaystyle \frac{C_{\theta }(u,v)}{u}}=v+\theta {\displaystyle \frac{u-v}{u}}g\left(u\right)g\left(v\right)$.
  The inequality (28) therefore becomes $\theta g\left(v\right)\left[{\displaystyle \frac{v}{u}}g\left(u\right)+(u-v)g^{\prime }\left(u\right)\right]⩽0$ by rearranging the terms. Furthermore, from $\left(27\right)$, we obtain $\left[{\displaystyle \frac{1}{u}}g\left(u\right)-g^{\prime }\left(u\right)\right]v+u{g}^{\prime }\left(u\right)⩽0$.
  Let $\mathcal{A}\left(v\right)=\left[{\displaystyle \frac{1}{u}}g\left(u\right)-g^{\prime }\left(u\right)\right]v+u{g}^{\prime }\left(u\right)$ (an affine function with respect to v).
  If ${\displaystyle \frac{1}{u}}g\left(u\right)-g^{\prime }\left(u\right)⩾0$, this implies that $\mathcal{A}$ is increasing and, consequently, $\mathcal{A}$ reaches its maximum at $v=1$.
  Hence, $\mathcal{A}\left(1\right)⩽0$ is equivalent to $g\left(u\right)⩽u(1-u)g^{\prime }\left(u\right)$ for almost all u.
  If ${\displaystyle \frac{1}{u}}g\left(u\right)-g^{\prime }\left(u\right)⩽0$, this implies that $\mathcal{A}$ is decreasing; hence, $\mathcal{A}$ reaches its maximum at $v=0$.
  Therefore, $\mathcal{A}\left(0\right)⩽0$ is equivalent to $g^{\prime }\left(u\right)⩽0$, which contradicts $g^{\prime }\left(u\right)⩾{\displaystyle \frac{1}{u}}g\left(u\right)⩾0,$ for almost all u.
  The proof is similar for $\mathbf{LTD}\left(\mathbf{X}\right|\mathbf{Y})$. Following the same steps as before, we find that

  $${\displaystyle \frac{\partial C_{\theta }(u,v)}{\partial v}}⩽{\displaystyle \frac{C_{\theta }(u,v)}{v}},\mathrm{for}\mathrm{almost}\mathrm{all}v,$$

  which leads to $\left[{\displaystyle \frac{1}{v}}g\left(v\right)-g^{\prime }\left(v\right)\right]u+v{g}^{\prime }\left(v\right)⩾0$.
  Let $\mathcal{B}\left(u\right)=\left[{\displaystyle \frac{1}{v}}g\left(v\right)-g^{\prime }\left(v\right)\right]u+v{g}^{\prime }\left(v\right)$.
  If ${\displaystyle \frac{1}{v}}g\left(v\right)-g^{\prime }\left(v\right)⩾0$, which implies that $\mathcal{B}$ is increasing, then $\mathcal{B}$ reaches its minimum at $u=0$.
  Then, $\mathcal{B}\left(0\right)⩾0$ is equivalent to $g\left(v\right)⩽v(1-v)g^{\prime }\left(v\right),$ for almost all v.
  If ${\displaystyle \frac{1}{v}}g\left(v\right)-g^{\prime }\left(v\right)⩽0$, then $\mathcal{B}$ is decreasing and, consequently, $\mathcal{B}$ reaches its minimum at $u=1$.
  Thus, $\mathcal{B}\left(1\right)⩾0$ is equivalent to $g\left(v\right)⩾v(1-v)g^{\prime }\left(v\right),$ for almost all v.
• $\mathbf{RTI}\left(\mathbf{Y}\right|\mathbf{X})$ holds if and only if, for any $v\in I$,

  $${\displaystyle \frac{\partial C_{\theta }(u,v)}{\partial u}}⩾{\displaystyle \frac{v-C_{\theta }(u,v)}{1-u}},\mathrm{for}\mathrm{almost}\mathrm{all}u,$$

  which leads to $\left[{\displaystyle \frac{1}{1-u}}g\left(u\right)+g^{\prime }\left(u\right)\right]v-{\displaystyle \frac{1}{1-u}}g\left(u\right)-u{g}^{\prime }\left(u\right)⩽0$.
  Let $\mathcal{T}\left(v\right)=\left[{\displaystyle \frac{1}{1-u}}g\left(u\right)+g^{\prime }\left(u\right)\right]v-{\displaystyle \frac{1}{1-u}}g\left(u\right)-u{g}^{\prime }\left(u\right)$.
  If ${\displaystyle \frac{1}{1-u}}g\left(u\right)+g^{\prime }\left(u\right)⩽0$, then $\mathcal{T}$ reaches its maximum at $v=0$. $\mathcal{T}\left(0\right)⩽0$ is equivalent to $g\left(u\right)⩾u(u-1)g^{\prime }\left(u\right),$ for almost all u.
  If ${\displaystyle \frac{1}{1-u}}g\left(u\right)+g^{\prime }\left(u\right)⩾0$, then $\mathcal{T}$ reaches its maximum at $v=1$.
  Then, $\mathcal{T}\left(1\right)⩽0$ is equivalent to $g^{\prime }\left(u\right)⩽0,$ for almost all u (where $\mathcal{T}\left(1\right)=(1-u)g^{\prime }\left(u\right)$).
  $\mathbf{RTI}\left(\mathbf{X}\right|\mathbf{Y})$ can be proved using similar arguments.
• $\mathbf{SI}\left(\mathbf{Y}\right|\mathbf{X})$ holds if and only if, for any $v\in I$, $C(u,v)$ is a concave function of u.
  We have

  $${\displaystyle \frac{{\partial }^2{C}_{\theta }(u,v)}{\partial u^2}}=2\theta g^{\prime }\left(u\right)g\left(v\right)+\theta (u-v)g^{″}\left(u\right)g\left(v\right),\theta >0$$
  Then, ${\displaystyle \frac{{\partial }^2{C}_{\theta }(u,v)}{\partial u^2}}⩽0$ implies that, for any $v\in I$, $-g^{″}\left(u\right)v+2g^{\prime }\left(u\right)+u{g}^{″}\left(u\right)⩽0.$
  Let $\mathcal{F}\left(v\right)=-g^{″}\left(u\right)v+2g^{\prime }\left(u\right)+u{g}^{″}\left(u\right)$. As $g^{″}\left(u\right)⩽0$,
  this means that $\mathcal{F}$ is increasing; i.e., $\mathcal{F}$ reaches its maximum at $v=1$.
  Hence, $\mathcal{F}\left(1\right)⩽0$ is equivalent to $g^{\prime }\left(u\right)⩽{\displaystyle \frac{1-u}{2}}{g}^{″}\left(u\right)$, for almost all $u\in I$.
  The same reasoning applies to $\mathbf{SI}\left(\mathbf{Y}\right|\mathbf{X})$, and the desired result follows. □

4.4.3. Tail Dependence

Tail dependence quantifies the joint likelihood of extreme events in a bivariate distribution’s tails (see [3]).

Let X and Y be continuous random variables with respective distribution functions F and G. The upper and lower tail dependence coefficients (${\lambda }_U$ and ${\lambda }_L$, respectively) are given by

$${\lambda }_U=\underset{u\to 1^{-}}{lim}P\left[Y>G^{(-1)}\left(t\right)|X>F^{(-1)}\left(t\right)\right],$$

and

$${\lambda }_L=\underset{u\to 0^{+}}{lim}P\left[Y⩽G^{(-1)}\left(t\right)|X⩽F^{(-1)}\left(t\right)\right].$$

According to ([16], Theorem 5.4.2), if the limits exist, the upper and lower tail dependence coefficients ${\lambda }_U$ and ${\lambda }_L$ can also be characterized in terms of the copula associated with the joint distribution of $(X,Y)$. Let $C(u,v)$ denote the copula of the pair $(X,Y)$. Then,

$${\lambda }_U=\underset{u\to 1^{-}}{lim}2-{\displaystyle \frac{1-C_{\theta }(u,u)}{1-u}},$$

and

$${\lambda }_L=\underset{u\to 0^{+}}{lim}{\displaystyle \frac{C_{\theta }(u,u)}{u}}.$$

Proposition 5.

The copula $C_{\theta }$ of the form (12) has a total tail independence; i.e.,

$${\lambda }_U\left(C_{\theta }\right)={\lambda }_L\left(C_{\theta }\right)=0.$$

Proof. 

It is immediate to verify that $C_{\theta }(u,u)=u^2$, where $C_{\theta }$ is of the form (12).

• Hence,

$${\lambda }_U=\underset{u\to 1^{-}}{lim}2-{\displaystyle \frac{1-u^2}{1-u}}=\underset{u\to 1^{-}}{lim}2-1-u=0,$$

and

$${\lambda }_L=\underset{u\to 0^{+}}{lim}{\displaystyle \frac{u^2}{u}}=\underset{u\to 0^{+}}{lim}u=0. \square$$

(29)

Remark 4.

The tail independence observed in the proposed copula family is a direct consequence of the structural properties of the perturbation term. Specifically, the conditions $f\left(0\right)=f\left(1\right)=g\left(0\right)=g\left(1\right)=0$ ensure that the perturbation vanishes on the boundaries of the unit square, causing the copula to reduce to the independence copula Π in the corners. This behavior naturally leads to tail independence, a feature consistent with the construction’s underlying framework.

5. Asymmetry Analysis of Our New Class of Family of Copulas

In this section, we analyze the asymmetry of copulas belonging to our new class of families through the relationships between an asymmetry measure (non-exchangeability), denoted by ${\mu }_{\infty }$, and the four convex concordance measures $\rho ,\varphi ,\gamma$, and $\beta$, recently introduced by Bukovšek et al. [27].

The topic of asymmetry or non-exchangeability of copulas has recently received considerable attention and plays a prominent role in current research [28,29,30,31].

A function $\nu :\mathcal{C}\to [0,\infty [$ is called a measure of asymmetry (or non-exchangeability) for a copula C if it satisfies the following properties:

(A₁)

There exists $M\in R^{+}$ such that, for all $C\in \mathcal{C}$, we have $\nu \left(C\right)⩽M$;

(A₂)

$\nu \left(C\right)=0$ if and only if $C=\pi \left(C\right)$ (C is symmetric);

(A₃)

$\nu \left(C\right)=\nu \left(\pi \right(C\left)\right)=\nu \left(\varphi \right(C\left)\right),\mathrm{for}\mathrm{all}C\in \mathcal{C}$;

(A₄)

If ${\left(C_n\right)}_{n\in \mathbb{N}}$ and C are in $\mathcal{C}$, and if $(C_n{)}_{n\in \mathbb{N}}$ converges uniformly to C, then $\nu (C_n{)}_{n\in \mathbb{N}}$ converges to $\nu \left(C\right)$.

As an example, a broad class of asymmetry measures was proposed by Nelsen [32] (for a complementary reference, see [33]).

Let $d_p$ denote the classical $L_p$ distance on $\mathcal{C}$ for $p\in [1,+\infty ]$. For all $C_1,C_2\in \mathcal{C}$, we have the following.

• For $p\in [1,+\infty [$:

  $$d_p(C_1,C_2)={\left({\int }_{I^2}{|C_1(u,v)-C_2(u,v)|}^{p}dudv\right)}^{{\displaystyle \frac{1}{p}}}.$$
• For $p=+\infty$:

$$d_{+\infty }(C_1,C_2)=ma{x}_{(u,v)\in I^2}|C_1(u,v)-C_2(u,v)|.$$

The measure ${\nu }_p:\mathcal{C}\to {\mathbb{R}}^{+}$ is then defined as ${\nu }_p\left(C\right)=d_p(C,\pi \left(C\right))$ and can be normalized using ${\nu }_{\infty }$ (see [34]).

Property $\left(A_4\right)$ implies the continuity of $\nu$ with respect to the topology induced by the $L_{\infty }$ norm on $\mathcal{C}$ and the Euclidean norm on $\mathbb{R}$. The compactness of $\mathcal{C}$ therefore guarantees the existence of a maximum for $\nu$, which justifies the following result.

Proposition 6.

Let ${\nu }_p:C\to R^{+}$ for $p\in [1,+\infty ]$ be a measure of asymmetry. Then, there exists a constant $M_{{\nu }_p}\in R^{+}$ and a copula $C_{{\nu }_p}\in \mathcal{C}$ such that ${\nu }_p\left(C_{{\nu }_p}\right)=M_{{\nu }_p}$ and ${\nu }_p\left(C_{{\nu }_p}\right)⩾{\nu }_p\left(C\right)$ for all copulas $C\ne C_{{\nu }_p}$.

Proof. 

According to the result ([32], Lemme 2.1), for any copula C and any $(u,v)\in I^2$, we have

$$|C(u,v)-\pi \left(C\right)(u,v)|⩽d^{*}(u,v),\mathrm{where}{d}^{*}(u,v)=min\{u,v,1-u,1-v,|v-u\left|\right\}.$$

By integrating both sides of this inequality over the unit square we obtain, for all $p\in [1,+\infty [$,

$${\nu }_p\left(C\right)⩽{\left({\displaystyle \frac{2\times 3^{-p}}{(p+1)(p+2)}}\right)}^{{\displaystyle \frac{1}{p}}}. \square$$

Note that ${\nu }_p⩽{\displaystyle \frac{1}{3}}$ and $\underset{p\to +\infty }{lim}{\left({\displaystyle \frac{2\times 3^{-p}}{(p+1)(p+2)}}\right)}^{{\displaystyle \frac{1}{p}}}={\displaystyle \frac{1}{3}}.$

Now, choose $(a,b)\in I^2,\alpha \in I$ such that $0⩽\alpha ⩽d^{*}(a,b)$. From the convexity of $\mathcal{C}$, it readily follows that there exists a copula $C\in \mathcal{C}$ satisfying

$$C(a,b)-C(b,a)=\alpha .$$

(30)

We adopt the notation $x^{+}:=max(0,x)$ to denote the positive part of x.

Theorem 7.

For each $\alpha \in [0,d^{*}(a,b)]$, the set of copulas satisfying (30) admits extremal elements, ${C_L}_{\alpha }^{(a,b)}$ and ${C_U}_{\alpha }^{(a,b)}$; i.e., for every copula C satisfying (30), we have

$${C_L}_{\alpha }^{(a,b)}⩽C⩽{C_U}_{\alpha }^{(a,b)},$$

(31)

where

$${C_L}_{\alpha }^{(a,b)}=max\{W(u,v),min\{c_1+{(u-a)}^{+}+{(v-b)}^{+}\}\}$$

(32)

and

$${C_U}_{\alpha }^{(a,b)}=min\{M(u,v),max\{c_2+{(u-b)}^{+}+{(v-a)}^{+}\}\},$$

(33)

with $c_1:=c_{1,\alpha }^{(a,b)}=W(a,b)+\alpha$ and $c_2:=c_{2,\alpha }^{(a,b)}=M(a,b)-\alpha$.

Proof. 

• If $u⩾a$, by the 1-increasing property, we have $C(u,v)-C(a,v)⩾0$.
  If $u<a$, by the 1-Lipschitz property, we obtain

$$C(u,v)-C(a,v)⩾u-a.$$

Hence, it follows that

$$C(u,v)-C(a,v)⩾{(u-a)}^{+}.$$

(34)

Similarly, according to the same properties, it follows that:

If $v⩾b$, then $C(a,v)-C(a,b)⩾0$;

If $v<b$, then $C(a,v)-C(a,b)⩾v-b$,

• and, consequently,

$$C(a,v)-C(a,b)⩾{(v-b)}^{+}.$$

(35)

From (34) and (35), we obtain

$${(u-a)}^{+}+{(v-b)}^{+}⩽C(u,v)-C(a,b).$$

By (30) and as $W(a,b)=W(b,a)⩽C(b,a)$, we have

$$c_1+{(u-a)}^{+}+{(v-b)}^{+}⩽C(u,v)\mathrm{with}{c}_1=W(a,b)+\alpha .$$

In a manner analogous to the previous one, we find that

$$\left\{\begin{array}{ccc}C(u,v)-C(b,v)⩽u-b,\hfill & if& u>b;\hfill \\ C(u,v)-C(b,v)⩽0,\hfill & if& u⩽b,\hfill \end{array}\right.and\left\{\begin{array}{ccc}C(b,v)-C(b,a)⩽v-a,\hfill & if& v>a;\hfill \\ C(b,v)-C(b,a)⩽0,\hfill & if& v⩽a.\hfill \end{array}\right.$$

This leads us to

$$C(u,v)-C(b,a)⩽{(u-b)}^{+}+{(v-a)}^{+}$$

and, hence, $C(u,v)⩽c_2+{(u-b)}^{+}+{(v-a)}^{+}$, where $c_2=M(a,b)-\alpha$.

• Incorporating the Fréchet–Hoeffding bounds completes the proof. □

We call (32) and (33) the local Fréchet–Hoeffding bounds.

An alternative proof, structured around five distinct cases according to the regions defined by the scatterplots, is developed in ([27], Theorem 2).

The calculation of the concordance measures for the local Fréchet–Hoeffding bounds, defined by (32) and (33), respectively, requires evaluation of the concordance function Q (given by (17)) applied to these two copulas, as well as to $W,\Pi$, and M. The synthesis of the graphical observations (Figure 3) and the preliminary calculations then reveals that

$$\begin{array}{ccc}\hfill Q(A,{C_L}_{\alpha }^{(a,b)})& =& 4{\int }_0^{1}A(u,v)d{C_L}_{\alpha }^{(a,b)}-1\hfill \\ & =& 4{\int }_0^{a-c_1}A(u,1-u)du+4{\int }_{a-c_1}^{a}A(u,a+b-c_1-u)du\hfill \\ & +& 4{\int }_a^{1-b+c_1}A(u,1+c_1-u)du+4{\int }_{1-b+c_1}^{1}A(u,1-u)du\hfill \end{array}$$

(36)

and

$$\begin{array}{ccc}\hfill Q(A,{C_U}_{\alpha }^{(a,b)})& =& 4{\int }_0^{1}A(u,v)d{C_U}_{\alpha }^{(a,b)}-1\hfill \\ & =& 4{\int }_0^{c_2}A(u,u)du+4{\int }_{c_2}^{b}A(u,u+a-c_2)du\hfill \\ & +& 4{\int }_b^{a+b-c_2}A(u,u-b+c_2)du+4{\int }_{a+b-c_2}^{1}A(u,u)du.\hfill \end{array}$$

(37)

In accordance with ([27], Properties 4 and 5), through which the authors derive formulas (36) and (37) for every $A\in \{W,\Pi ,M\}$, the results obtained are symmetric with respect to both the main diagonal and the counter-diagonal as follows.

Let $(a,b)\in I,\alpha \in [0,min\{a,b,1-a,1-b\left\}\right]$. Then, for all $A\in \{W,\Pi ,M\}$,

$$Q(A,{C_L}_{\alpha }^{(a,b)})=Q(A,{C_L}_{\alpha }^{(b,a)})=Q(A,{C_L}_{\alpha }^{(1-a,1-b)})$$

(38)

and

$$Q(A,{C_U}_{\alpha }^{(a,b)})=Q(A,{C_U}_{\alpha }^{(b,a)})=Q(A,{C_U}_{\alpha }^{(1-a,1-b)}).$$

(39)

Let $C\in \mathcal{C}$ be any copula with ${\nu }_{\infty }\left(C\right)=m$, and assume there exists a pair $(a_0,b_0)\in I$ such that

$$|C(a_0,b_0)-C(b_0,a_0)|=m.$$

Then, $C(a_0,b_0)-C(b_0,a_0)=m⩾0$ (if $C(a_0,b_0)-C(b_0,a_0)⩽0$, we replace C with $\pi \left(C\right)$).

By virtue of Theorem 7, it follows that

$${C_L}_m^{(a_0,b_0)}⩽C⩽{C_U}_m^{(a_0,b_0)}.$$

As an immediate consequence of monotonicity, we have $\forall \kappa \in \{\rho ,\varphi ,\gamma \}$,

$$\kappa \left({C_L}_m^{(a_0,b_0)}\right)⩽\kappa \left(C\right)⩽\kappa \left({C_U}_m^{(a_0,b_0)}\right).$$

Given the symmetries discussed in (38) and (39), we fix m. The study can therefore be restricted to the set $\mathcal{S}=\{(a,b)\in [0,1]|a⩽b,b⩽1-a,d^{*}(a,b)⩾m\}$ without loss of generality.

The analysis is therefore limited to the triangle $\Delta EFG:=\mathcal{S}=\left\{\right(a,b)\in [0,1\left]\right|a⩾m,b-a⩾m,b+a⩽1-2m\}$ with the vertices $E(m,2m),F({\displaystyle \frac{1-m}{2}},{\displaystyle \frac{1+m}{2}})$ and $G(m,1-m)$.

As a result, we have

$$\underset{(a_0,b_0)\in \mathcal{S}}{min}\kappa \left({C_L}_m^{(a_0,b_0)}\right)⩽\kappa \left(C\right)⩽\underset{(a_0,b_0)\in \mathcal{S}}{max}\kappa \left({C_U}_m^{(a_0,b_0)}\right).$$

Following ([27], Corollary 22), the minimum value of $\kappa \left({C_L}_m^{(a_0,b_0)}\right)$ and the maximum value of $\kappa \left({C_U}_m^{(a_0,b_0)}\right)$ on the set $\mathcal{S}$ are attained at the same points, denoted $G(m,1-m)$ and $E(m,2m)$, respectively. The preceding developments make it possible to establish a link between the measure ${\nu }_{\infty }$ and each measure $\kappa \in \{\rho ,\varphi ,\gamma ,\beta \}$.

Theorem 8.

Let $C\in \mathcal{C}$ be any copula with ${\nu }_{\infty }\left(C\right)=m$. Then,

$$\kappa \left(C\right)\in [\kappa \left({C_L}_m^{(m,1-m)}\right),\kappa \left({C_U}_m^{(m,2m)}\right)],\mathit{for}all\kappa \in \{\rho ,\varphi ,\gamma ,\beta \}.$$

• In particular:

• For $\kappa =\rho$, we have $\rho \left(C\right)\in \left[f\right(m),g(m\left)\right]$, where

  $$f\left(m\right)=12m^3-1andg\left(m\right)=1-36m^3.$$
• For $\kappa =\varphi$, we have $\varphi \left(C\right)\in \left[f\right(m),g(m\left)\right]$, where

  $$f\left(m\right)=\left\{\begin{array}{ccc}-{\displaystyle \frac{1}{2}},\hfill & if& 0⩽m⩽{\displaystyle \frac{1}{4}};\hfill \\ 24m^2-12m+1,\hfill & if& {\displaystyle \frac{1}{4}}⩽m⩽{\displaystyle \frac{1}{3}}.\hfill \end{array}\right.andg\left(m\right)=1-12m^2.$$
• For $\kappa =\gamma$, we have $\gamma \left(C\right)\in \left[f\right(m),g(m\left)\right]$, where

  $$f\left(m\right)=\left\{\begin{array}{ccc}4m^2-1,\hfill & if& 0⩽m⩽{\displaystyle \frac{1}{4}};\hfill \\ 20m^2-8m,\hfill & if& {\displaystyle \frac{1}{4}}⩽m⩽{\displaystyle \frac{1}{3}},\hfill \end{array}\right.$$

  and

  $$g\left(m\right)=\left\{\begin{array}{ccc}1-8m^2,\hfill & if& 0⩽m⩽{\displaystyle \frac{1}{6}};\hfill \\ -44m^2+12m,\hfill & if& {\displaystyle \frac{1}{6}}⩽m⩽{\displaystyle \frac{1}{5}};\hfill \\ -19m^2+2m+1,\hfill & if& {\displaystyle \frac{1}{5}}⩽m⩽{\displaystyle \frac{1}{4}};\hfill \\ -3m^2-6m+2,\hfill & if& {\displaystyle \frac{1}{4}}⩽m⩽{\displaystyle \frac{1}{3}}.\hfill \end{array}\right.$$
• For $\kappa =\beta$, we have $\beta \left(C\right)\in \left[f\right(m),g(m\left)\right]$, where

  $$f\left(m\right)=\left\{\begin{array}{ccc}-1,\hfill & if& 0⩽m⩽{\displaystyle \frac{1}{4}};\hfill \\ 8m-3,\hfill & if& {\displaystyle \frac{1}{4}}⩽m⩽{\displaystyle \frac{1}{3}},\hfill \end{array}\right.$$

  and

  $$g\left(m\right)=\left\{\begin{array}{ccc}1,\hfill & if& 0⩽m⩽{\displaystyle \frac{1}{6}};\hfill \\ -12m+3,\hfill & if& {\displaystyle \frac{1}{6}}⩽m⩽{\displaystyle \frac{1}{4}};\hfill \\ -4m+1,\hfill & if& {\displaystyle \frac{1}{4}}⩽m⩽{\displaystyle \frac{1}{3}}.\hfill \end{array}\right.$$

Note that, if the copula C takes the maximal possible asymmetry ${\nu }_{\infty }\left(C\right)={\displaystyle \frac{1}{3}}$, then

$\rho \left(C\right)\in [-{\displaystyle \frac{5}{9}},-{\displaystyle \frac{1}{3}}],\gamma \left(C\right)\in \left[-{\displaystyle \frac{4}{9}},-{\displaystyle \frac{1}{3}}\right],$ and $\varphi \left(C\right)=\beta \left(C\right)=-{\displaystyle \frac{1}{3}}$.

Regarding our new class of family of copulas, the use of inverses immediately yields the following result.

Corollary 2.

Let C be the copula satisfying property (6). Then:

If $\kappa =\rho$, then $0⩽{\nu }_{\infty }\left(C\right)⩽\sqrt[3]{{\displaystyle \frac{1}{36}}}$.

If $\kappa =\varphi$, then $0⩽{\nu }_{\infty }\left(C\right)⩽\sqrt{{\displaystyle \frac{1}{12}}}$.

If $\kappa =\gamma$, then $0⩽{\nu }_{\infty }\left(C\right)⩽{\displaystyle \frac{\sqrt{15}}{3}}-1$.

If $\kappa =\beta$, then $0⩽{\nu }_{\infty }\left(C\right)⩽{\displaystyle \frac{1}{4}}$.

Remark 5.

For all copulas $C\in \mathcal{C}$, if $\tau \left(C\right)\in \left[-{\displaystyle \frac{5}{9}},{\displaystyle \frac{1}{9}}\right]$, then C takes the maximal possible asymmetry ${\nu }_{\infty }\left(C\right)={\displaystyle \frac{1}{3}}$ (see [32]). Inductive inference based on the data from Figure 2 reveals that, for any copula C belonging to the parametric family given by (14), we have $0⩽{\nu }_{\infty }\left(C\right)⩽{\displaystyle \frac{1}{3}}.$

• For each copula of the form $C(u,v)=uv+P(u,v)$, where $P(u,v)=-P(v,u),\mathit{for}all(u,v)\in I^2$, we have $|C(u,v)-C(v,u)|=2\left|P\right|,\mathit{for}all(u,v)\in I^2$. Corollary 2 directly implies that $0⩽\left|P\right|⩽{\displaystyle \frac{1}{8}}$.

This appears to be a good indicator of the mildness of the perturbation.

6. Conclusions and Directions for Further Research

This article defines and studies a class of parametric antisymmetric copula families and establishes their dependence and asymmetry properties. The methodology developed for analysis of the concordance and dependence measures relies on a family of convex weak concordance measures, allowing for inference of a nonlinear dependence structure characterizing the considered copula class. This, in turn, provides an optimal modeling framework for various phenomena in which weak dependencies reveal critical nonlinear structures.

In line with our previous approach, our future work will involve characterizing and analyzing a family of copulas arising from an anti-radially symmetric perturbation of the independence copula. It is worth noting that this perturbative approach is extensible to the multivariate case, which will be addressed separately in future work.

Another research direction is to identify and study another class of copulas, whose symmetrized part is Archimedean.