Constructions of Multivariate Copulas

Abstract

In this chapter, several general methods of constructions of multivariate copulas are presented, which are generalizations of some existing constructions in bivariate copulas. Dependence properties of new families are explored and examples are given for illustration of our results.

1 Introduction

In recent years, copulas are hot topics in probability and statistics. By Sklar theorem [16], the importance of copulas comes from two aspects, (1) describing dependence properties of random variables, such as Joe [6], Nelsen [11], Siburg [15], Tasena [17], Shan [14], Wei [20]; and (2) constructing the joint distributions of random variables. In the second direction, there are many papers devoting to the constructions of bivariate copulas, such as Rodríguez-Lallena [12], Kim [7], Durante [4], Mesiar [9], Aguilo [1], Mesiar [10], but few of constructions of multivariate copulas, such as Liebscher [8], Durante [3].

In this paper, we discussed several general methods of constructing multivariate copulas, which are generalizations of some bivariate results. The paper is organized as follows: In Sect. 2, we introduce some necessary definitions and existing results. Several general methods for constructing multivariate copulas are provided in Sect. 3 and their dependence properties are discussed in Sect. 4. Finally, two examples are given in Sect. 5.

2 Definitions and Existing Results

A function $C: I^n\rightarrow I$ is called an n-copula [11], where $I=[0, 1]$, if C satisfies the following properties:

• (i) C is grounded, i.e., for any $\mathbf u =(u_1, \cdots , u_n)^\prime \in I^n$, if at least one $u_i=0$, then $C(\mathbf u )=0$,

• (ii) One-dimensional marginals of C are uniformly distributed, i.e., for any $u_i\in I$, $i=1, \cdots , n$,

  $$C(1, \cdots , 1, u_i, 1, \cdots , 1)=u_i,$$

• (iii) C is n-increasing, i.e., for any $\mathbf u$, $\mathbf v \in I^n$ such that $\mathbf u \le \mathbf v$, we have

  $$V_C([\mathbf u , \mathbf v ])=\sum sgn(\mathbf a )C(\mathbf a )\ge 0,$$

where the sum is taken over all vertices $\mathbf a$ of the n-box $[\mathbf u , \mathbf v ]=[u_1, v_1]\times \cdots \times [u_n, v_n]$, and

$$\begin{aligned}sgn(\mathbf a )= {\left\{ \begin{array}{ll} \ \ 1, &{} \text {if}\ a_i=u_i\ \text {for an even number of}\ i^\prime s, \\ -1, &{} \text {if}\ a_i=u_i\ \text {for an odd number of}\ i^\prime s. \end{array}\right. } \end{aligned}$$

Equivalently,

$$V_C([\mathbf u , \mathbf v ])=\varDelta ^\mathbf{v }_\mathbf{u }C(\mathbf t )=\varDelta ^{v_n}_{u_n}\cdots \varDelta ^{v_1}_{u_1}C(\mathbf t ),$$

where $\varDelta ^{v_k}_{u_k}C(\mathbf t )=C(t_1, \cdots , t_{k-1}, v_k, t_{k+1},\cdots , t_n)-C(t_1, \cdots , t_{k-1}, u_k, t_{k+1},\cdots , t_n)$, $k=1,\cdots ,n$.

Note that above three conditions ensure that the range of C is I. By Sklar’s theorem [16], any n random variables $X_1, \cdots , X_n$ can be connected by an n-copula via the equation

$$F(x_1, \cdots , x_n)=C(F_1(x_1), \cdots , F_n(x_n)),$$

where F is the joint distribution function of ${X_1}, \cdots , X_n$, $F_i$ is the marginal distribution functions of $X_i$, $i=1, \cdots , n$. In addition, if ${X_1}, \cdots , X_n$ are continuous, then the copula C is unique.

There are three important functions for n-copulas defined respectively by

$$M_n(\mathbf u )=\min \{u_1, \cdots , u_n\},$$

$$\varPi _n(\mathbf u )=\underset{i=1}{\overset{n}{\prod }}u_i,$$

and

$$W_n(\mathbf u )=\max \{u_1+\cdots +u_n-n+1, 0\},$$

for all $\mathbf u \in I^n$. Functions $M_n$ and $\varPi _n$ are n-copulas for all $n\ge 2$, but $W_n$ is not an n-copula for any $n\ge 3$. $M_n$ and $W_n$ are called the Fréchert-Hoeffding upper bound and lower bound of n-copulas respectively since for any n-copula C, we have $W_n\le C \le M_n$.

Let $H: I^n\rightarrow \mathbb {R}$ be a function. The functions $H_{i_1i_2\cdots i_k}: I^k\rightarrow \mathbb {R}$ are called k-dimensional marginals of H defined by

$$H_{i_1i_2\cdots i_k}(u_{i_1}, \cdots , u_{i_k})=H(v_1, \cdots ,\ v_n),$$

where $v_j=u_{i_l}$ if $j=i_l$ for some $l=1, 2, \cdots , k$, otherwise, $v_j=1$.

Any n-copula C defines a function $\overline{C}: I^n\rightarrow I$ by

$$\begin{aligned} \overline{C}(\mathbf u )=1+ \underset{k=1}{\overset{n}{\sum }}(-1)^k\underset{1\le i_1<\cdots < i_k\le n}{\sum }C_{i_1i_2\cdots i_k}(u_{i_1}, \cdots , u_{i_k}). \end{aligned}$$

(1)

It is called the survival functionof C. For more details about copulas theory, see Nelsen’s book [11].

Now let’s recall some existing results. In 2004, Rodríguez-Lallena and Úbeda-Flores [12] considered the following family of bivariate copulas,

$$\begin{aligned} C_\theta (u, v)=uv+\theta f(u)g(v), \end{aligned}$$

(2)

where $f, g: [0, 1]\rightarrow \mathbb {R}$ are two functions, $\theta \in \mathbb {R}$ is a parameter. This family is a generalization of the well-known bivariate Farlie-Gumble-Morgenstern (or FGM, for short) family,

$$C_\theta (u, v)=uv+\theta uv(1-u)(1-v),$$

where $u, v \in [0, 1]$ and $\theta \in [-1, 1]$. In 2011, Kim et al. [7] extended Rodríguez-Lallena and Úbeda-Flores’s work to the family,

$$\begin{aligned} C(u, v)=C^*(u, v)+ \theta f(u)g(v), \end{aligned}$$

(3)

where $C^*$ is a known bivariate copula, $f, g: [0, 1]\rightarrow \mathbb {R}$ are two functions, $\theta$ is a parameter. In 2013 and 2015, Durante et al. [4] and Mesiar et al. [10] considered more general cases,

$$\begin{aligned} C(u, v)=C^*(u, v)+ H(u, v), \end{aligned}$$

(4)

where $C^*$ is a known bivariate copula, $H: [0, 1]\times [0, 1]\rightarrow \mathbb {R}$ is a function.

3 Constructions of Multivariate Copulas

The constructions of all above results are adding some perturbation functions to a given bivariate copula. In fact, any n-copula C can be represented by a perturbation of the independent copula $\varPi _n$ [19]. Based on this idea, we are going to extend these bivariate results to multivariate cases.

Firstly, for any given n-copula $C^*: I^n\rightarrow [0, 1]$, we consider the construction,

$$\begin{aligned} C(u_1, u_2, \cdots , u_n)=C^*(u_1, u_2, \cdots , u_n)+H(u_1, u_2, \cdots , u_n), \end{aligned}$$

(5)

where $H: I^n\rightarrow \mathbb {R}$ is a function, called a perturbation function. C is called a perturbation of $C^*$ by H.

Theorem 1

Let $C^*$ be an n-copula, $H: I^n\rightarrow \mathbb {R}$ be a function. C is defined by (5). Then C is an n-copula if and only if H satisfies the following three conditions,

• (i) $H(0, u_2, \cdots , u_n)=\cdots =H(u_1, \cdots , u_{n-1}, 0)=0$ for all $(u_1, \cdots , u_n)\in I^n$,

• (ii) There are $1\le i<j\le n$ such that

  $$H(u_1, \cdots , u_{i-1}, 1, u_{i+1}, \cdots , u_n)=H(u_1, \cdots , u_{j-1}, 1, u_{j+1}, \cdots , u_n)=0,$$

• (iii) $V_{C^*}([\mathbf u , \mathbf v ])+V_H([\mathbf u , \mathbf v ])\ge 0$ for all n-box $[\mathbf u , \mathbf v ]$ in $I^n$.

Proof

The conditions (i) and (ii) ensure that C is grounded, and its one-dimensional marginals are uniform distributed, respectively. The n-increasing property of C is guaranteed by the condition (iii).       $\square$

Next we provide a necessary and sufficient condition on H under which C defined by (5) is an absolutely continuous n-copula.

Theorem 2

Let $C^*$ be an absolutely continuous n-copula with the density $c^*$, $H: I^n\rightarrow \mathbb {R}$ be a non-zero absolutely continuous function with the Radon-Nikodym derivative h with respect to the Lebesgue measure on $I^n$. C is defined by (5) is an absolutely continuous n-copula if and only if H satisfies the following conditions.

• (i) $H(0, u_2, \cdots , u_n)=\cdots =H(u_1, \cdots , u_{n-1}, 0)=0$ for all $(u_i, \cdots , u_n)\in I^n$,

• (ii) There are $1\le i<j\le n$ such that

  $$H(u_1, \cdots , u_{i-1}, 1, u_{i+1}, \cdots , u_n)=H(u_1, \cdots , u_{j-1}, 1, u_{j+1}, \cdots , u_n)=0,$$

• (iii) $c^*+h\ge 0$ almost surely.

Proof

Firstly, the boundary conditions of copulas are ensured by the condition (i) and (ii).

Next, we show that the condition (iii) is equivalent to the n-increasing property of C. On the one hand, suppose that C is n-increasing. If $c^*+h$ is not nonnegative almost surely, then there exist $\mathbf u <\mathbf v \in I^n$ such that $c^*+h< 0$ on $[\mathbf u , \mathbf v ]$. Note that $V_C([\mathbf u , \mathbf v ])=\int _{[\mathbf u , \mathbf v ]}(c^*+h)(\mathbf t )d\mathbf t$. So $V_C([\mathbf u , \mathbf v ])<0$. It contradicts the n-increasing property of C. On the other hand, if $c^*+h\ge 0$ almost surely, we must have $V_C([\mathbf u , \mathbf v ])\ge 0$ for all $\mathbf u$, $\mathbf v \in I^n$ with $\mathbf u \le \mathbf v$.       $\square$

Now let’s consider a special case of (5) as follows, which are multivariate extensions of the result in [7].

$$\begin{aligned} C(u_1, u_2, \cdots , u_n)=C^*(u_1, u_2, \cdots , u_n)+ \overset{n}{\underset{i=1}{\prod }}f_i(u_i), \end{aligned}$$

(6)

where $C^*$ is an n-copula, $f_i:[0,1]\rightarrow \mathbb {R}$ is a function, $i=1, 2, \cdots , n$.

The following theorem give us a sufficient condition under which C defined by (6) is an n-copula.

Theorem 3

Let $C^*$ be an n-copula, $f_i:[0,1]\rightarrow \mathbb {R}$ be a function, $i=1, 2, \cdots , n$. $C:[0, 1]^n\rightarrow \mathbb {R}$ is defined by (6) is an n-copula if ${f_1}, \cdots , f_n$ satisfy the following conditions,

• (i) $f_1(0)=\cdots =f_n(0)=0$, and there exist at least two functions $f_i$ and $f_j$ such that $f_i(1)=f_j(1)=0$, $1\le i, j\le n$,

• (ii) $f_i$ is absolutely continuous,

• (iii) $\min (B)\ge \sup \left\{ - \dfrac{V_{C^*}([\mathbf u , \mathbf v ])}{\varDelta (\mathbf u , \mathbf v )}: \mathbf u , \mathbf v \in [0, 1]^n, \mathbf u <\mathbf v \right\}$, where $B=\{\alpha _{i_1}\cdots \alpha _{i_k}\beta _{j_1}\cdots \beta _{j_{n-k}}: 1\le k\le n, k\ is\ odd, i_1, \cdots , i_k\ and\ j_1, \cdots , j_{n-k}\ are\ pairwise\ distinct\}$, $\alpha _i=\inf \{f^{\prime }_i(u_i): u_i\in A_i\}<0$, $\beta _i=\sup \{f^{\prime }_i(u_i): u_i\in A_i\}>0$, $A_i=\{u_i \in [0, 1]: f^{\prime }(u_i)\ exists\}$, $i=1, \cdots , n$, and $\varDelta (\mathbf u , \mathbf v )=(v_1-u_1)\cdots (v_n-u_n)$.

Proof

Firstly, if there is $f_i=0$, then $C=C^*$ is an n-copula. So without loss of generality, we may assume that $f_i$ is non-zero, $i=1, \cdots , n$.

Since $C^*$ is an n-copula, C is grounded and its marginals are uniformly distributed if and only if C satisfies the above condition (i). Next we are going to show that C is n-increasing if C satisfies the conditions (ii) and (iii).

Suppose that C satisfies conditions (ii) and (iii). By Lemma 2.2 in [12], it holds that for any $\mathbf u$, $\mathbf v \in I^n$ with $\mathbf u <\mathbf v$,

$$\begin{aligned} \dfrac{(f_1(v_1)-f_1(u_1))\cdots (f_n(v_n)-f_n(u_n))}{(v_1-u_1)\cdots (v_n-u_n)}\ge -\dfrac{V_{C^*}([\mathbf u , \mathbf v ])}{\varDelta (\mathbf u , \mathbf v )}, \end{aligned}$$

i.e.,

$$V_{C}([\mathbf u , \mathbf v ])=V_{C^*}([\mathbf u , \mathbf v ])+(f_1(v_1)-f_1(u_1))\cdots (f_n(v_n)-f_n(u_n))\ge 0,$$

so C is n-increasing.       $\square$

Based on the construction (6), we introduce the following parametric families of n-copulas, which is a multivariate extension of (3).

$$\begin{aligned} C(u_1, u_2, \cdots , u_n)=C^*(u_1, u_2, \cdots , u_n)+ \theta \overset{n}{\underset{i=1}{\prod }}f_i(u_i), \end{aligned}$$

(7)

where $C^*$ is an n-copula, $f_i:[0,1]\rightarrow \mathbb {R}$ is a function, $i=1, 2, \cdots , n$, $\theta \in \mathbb {R}$.

Corollary 1

Let $C^*$ be an n-copula, $f_i:[0,1]\rightarrow \mathbb {R}$ be a function, $i=1, 2, \cdots , n$. $C:[0, 1]^n\rightarrow \mathbb {R}$ is defined by (6) is an n-copula if ${f_1}, \cdots , f_n$ and $\theta$ satisfy the following conditions,

• (i) $f_1(0)=\cdots =f_n(0)=0$, and there exist at least two functions $f_i$ and $f_j$ such that $f_i(1)=f_j(1)=0$, $1\le i, j\le n$,

• (ii) $f_i$ is absolutely continuous,

• (iii) $\sup \{- \dfrac{V_{C^*}([\mathbf u , \mathbf v ])}{\varDelta (\mathbf u , \mathbf v )}: \mathbf u , \mathbf v \in [0, 1]^n, \mathbf u<\mathbf v \}\dfrac{1}{\max (B^\prime )}\le \theta \le \sup \{- \dfrac{V_{C^*}([\mathbf u , \mathbf v ])}{\varDelta (\mathbf u , \mathbf v )}: \mathbf u , \mathbf v \in [0, 1]^n, \mathbf u <\mathbf v \}\dfrac{1}{\min (B)}$, where B is the same as Theorem 3, $B^\prime =\{\alpha _{i_1}\cdots \alpha _{i_k}\beta _{j_1}\cdots \beta _{j_{n-k}}: 1\le k\le n,\ k\ is\ even, i_1, \cdots , i_k\ and\ j_1, \cdots , j_{n-k}\ are\ pairwise\ distinct\}$, $\alpha _i=\inf \{f^{\prime }_i(u_i): u_i\in A_i\}<0$, $\beta _i=\sup \{f^{\prime }_i(u_i): u_i\in A_i\}>0$, $A_i=\{u_i \in [0, 1]: f^{\prime }(u_i)\ exists\}$, $i=1, \cdots , n$, and $\varDelta (\mathbf u , \mathbf v )=(v_1-u_1)\cdots (v_n-u_n)$.

Remark 1

Conditions in Theorem 3 and Corollary 1 are sufficient but may not be necessary. Consider the Fréchert-Hoeffding upper bound of n-copulas, $M_n(u_1, \cdots , u_n)=\min \{u_1, \cdots , u_n\}$. For any $\mathbf u , \mathbf v \in [0, 1]^n$ such that $\mathbf u <\mathbf v$, it can be shown that

$$V_{M_n}([\mathbf u , \mathbf v ])=\max \{\min \{v_1, \cdots , v_n\}-\max \{u_1, \cdots , u_n\}, 0\}.$$

Thus,

$$\sup \left\{ - \dfrac{V_{C^*}([\mathbf u , \mathbf v ])}{\varDelta (\mathbf u , \mathbf v )}: \mathbf u , \mathbf v \in [0, 1]^n, \mathbf u <\mathbf v \right\} =0.$$

So functions $f_1, \cdots , f_n$ that satisfy conditions in Theorem 3 or Corollary 1 for $M_n$ must be zero, i.e., $f_1=\cdots =f_n=0$.

Next we provide a stronger sufficient condition on $f_1, \cdots , f_n$ to ensure that C defined by (6) is an n-copula. Example 2.1 in [12] shows that the condition is not necessary.

Theorem 4

Let C be defined by (6). C is an n-copula if $f_1, \cdots , f_n$ satisfy the following conditions,

(i) $f_1(0)=\cdots =f_n(0)=0$, and there exist at least two functions $f_i$ and $f_j$ such that $f_i(1)=f_j(1)=0$, $1\le i, j\le n$,

(ii) $f_i$ satisfies the Lipschitz condition,

$$|f_i(v)-f_i(u)|\le M_i|v-u|,$$

for all $u, v \in I$, such that $M_i>0$, $i=1, \cdots , n$, and

$$\underset{1}{\overset{n}{\prod }} M_i\le \inf \left\{ \dfrac{V_{C^*}([\mathbf u , \mathbf v ])}{\varDelta (\mathbf u , \mathbf v )}: \mathbf u , \mathbf v \in [0, 1]^n, \mathbf u \le \mathbf v \right\} .$$

Proof

By the condition (i), C is grounded and one-dimensional marginals of C are uniformly distributed. For any $\mathbf u$, $\mathbf v \in I^n$ with $\mathbf u <\mathbf v$, by the condition (ii), we have

$$\begin{aligned} \begin{aligned} -\dfrac{(f_1(v_1)-f_1(u_1))\cdots (f_n(v_n)-f_n(u_n))}{(v_1-u_1)\cdots (v_n-u_n)}&\le \dfrac{|f_1(v_1)-f_1(u_1)|\cdots |f_n(v_n)-f_n(u_n)|}{|v_1-u_1|\cdots |v_n-u_n|}\\&\le \underset{1}{\overset{n}{\prod }} M_i\\&\le \inf \left\{ \dfrac{V_{C^*}([\mathbf u , \mathbf v ])}{\varDelta (\mathbf u , \mathbf v )}: \mathbf u , \mathbf v \in [0, 1]^n, \mathbf u \le \mathbf v \right\} . \end{aligned} \end{aligned}$$

So

$$\dfrac{(f_1(v_1)-f_1(u_1))\cdots (f_n(v_n)-f_n(u_n))}{(v_1-u_1)\cdots (v_n-u_n)}\ge \sup \left\{ - \dfrac{V_{C^*}([\mathbf u , \mathbf v ])}{\varDelta (\mathbf u , \mathbf v )}: \mathbf u , \mathbf v \in [0, 1]^n, \mathbf u \le \mathbf v \right\} .$$

Thus, as the proof of Theorem 3, C is n-increasing.       $\square$

4 Properties of New Families

In this section, we are going to study some non-parametric copula-based measures of multivariate association, some dependence concepts for copulas defined in Sect. 3 and some properties of those families.

Firstly, recall that the multivariate generalizations of Kendall’s tau, Spearman’s rho, and Blomqvist’s beta (see [13, 18] for details) are given by

$$\begin{aligned} \tau _n(C)=\dfrac{1}{2^{n-1}-1}\left[ 2^n\int _{I^n} C(\mathbf u )dC(\mathbf u )-1\right] , \end{aligned}$$

(8)

$$\begin{aligned} \rho _n(C)=\dfrac{n+1}{2^{n}-n-1}\left[ 2^{n-1} \left( \int _{I^n} C(\mathbf u )d\varPi _n(\mathbf u )+\int _{I^n} \varPi _n(\mathbf u )dC(\mathbf u )\right) -1\right] , \end{aligned}$$

(9)

$$\begin{aligned} \beta _n(C)=\dfrac{2^{n-1}\left[ C\left( \frac{1}{2}{} \mathbf 1 _n\right) +\overline{C}\left( \frac{1}{2}{} \mathbf 1 _n\right) \right] -1}{2^{n-1}-1}, \end{aligned}$$

(10)

where $\mathbf 1 _n$ is the vector $(1, \cdots , 1)^\prime \in \mathbb {R}^n$.

Theorem 5

Let C be an n-copula defined by (5), then the Kendall’s tau, Spearman’s rho, and Blomqvist’s beta of C are given by

$$\begin{aligned} \tau _n(C)=\tau _n(C^*)+\tau _n(H)+a_1, \end{aligned}$$

(11)

$$\begin{aligned} \rho _n(C)=\rho _n(C^*)+\rho _n(H)+\dfrac{n+1}{2^{n}-n-1}, \end{aligned}$$

(12)

$$\begin{aligned} \beta _n(C)=\beta _n(C^*)+\beta _n(H)+\dfrac{1-2^{n-1}}{2^n-1}, \end{aligned}$$

(13)

where $a_1=\dfrac{1}{2^{n-1}-1}\left[ 2^n\int _{I^n}C^*(\mathbf u )dH(\mathbf u )+2^n\int _{I^n}H(\mathbf u )dC^*(\mathbf u )+1\right]$.

Proof

Firstly, by the definition of $\tau _n$,

$$\begin{aligned} \begin{aligned} \tau _n(C)&=\dfrac{1}{2^{n-1}-1}\left[ 2^n\int _{I^n} C(\mathbf u )dC(\mathbf u )-1\right] \\&=\dfrac{1}{2^{n-1}-1}\left[ 2^n\int _{I^n} C^*(\mathbf u )+ H(\mathbf u )d(C^*(\mathbf u )+ H(\mathbf u ))-1\right] \\&=\dfrac{1}{2^{n-1}-1}[2^n\int _{I^n} C^*(\mathbf u )dC^*(\mathbf u )+2^n\int _{I^n}H(\mathbf u )dH(\mathbf u )\\&\ \ \ \ \ \ +2^n\int _{I^n}C^*(\mathbf u )dH(\mathbf u )+2^n\int _{I^n}H(\mathbf u )dC^*(\mathbf u )-1]\\&=\tau _n(C^*)+\tau _n(H)+a_1. \end{aligned} \end{aligned}$$

Secondly, by the definition of $\rho _n$,

$$\begin{aligned} \begin{aligned} \rho _n(C)&=\dfrac{n+1}{2^{n}-n-1}\left\{ 2^{n-1} \left[ \int _{I^n} C(\mathbf u )d\varPi _n(\mathbf u )+\int _{I^n} \varPi _n(\mathbf u )dC(\mathbf u )\right] -1\right\} \\&=\dfrac{n+1}{2^{n}-n-1}\left\{ 2^{n-1} \left[ \int _{I^n} C^*(\mathbf u )+ H(\mathbf u )d\varPi _n(\mathbf u )+\int _{I^n} \varPi _n(\mathbf u )d(C^*(\mathbf u )+ H(\mathbf u ))\right] -1\right\} \\&=\dfrac{n+1}{2^{n}-n-1}\{2^{n-1} [\int _{I^n} C^*(\mathbf u )d\varPi _n(\mathbf u )+\int _{I^n}H(\mathbf u )d\varPi _n(\mathbf u )+ \int _{I^n} \varPi _n(\mathbf u )dC^*(\mathbf u )\\&\ \ \ \ \ +\int _{I^n}\varPi _n(\mathbf u )dH(\mathbf u )]-1\}\\&=\rho _n(C^*)+\rho _n(H)+\dfrac{n+1}{2^{n}-n-1}. \end{aligned} \end{aligned}$$

Lastly, by the definition of survival functions, for any $\mathbf u \in I^n$,

$$\begin{aligned} \begin{aligned} \overline{C}(\mathbf u )&=1+ \underset{k=1}{\overset{n}{\sum }}(-1)^k\underset{1\le i_1<\cdots< i_k\le n}{\sum }C_{i_1i_2\cdots i_k}(u_{i_1}, \cdots , u_{i_k})\\&=1+ \underset{k=1}{\overset{n}{\sum }}(-1)^k\underset{1\le i_1<\cdots< i_k\le n}{\sum }\left[ C^*_{i_1i_2\cdots i_k}(u_{i_1}, \cdots , u_{i_k})+H_{i_1i_2\cdots i_k}(u_{i_1}, \cdots , u_{i_k})\right] \\&=1+ \underset{k=1}{\overset{n}{\sum }}(-1)^k\underset{1\le i_1<\cdots< i_k\le n}{\sum }C^*_{i_1i_2\cdots i_k}(u_{i_1}, \cdots , u_{i_k})\\&\ \ \ \ \ +1+ \underset{k=1}{\overset{n}{\sum }}(-1)^k\underset{1\le i_1<\cdots < i_k\le n}{\sum }H_{i_1i_2\cdots i_k}(u_{i_1}, \cdots , u_{i_k})-1\\&=\overline{C^*}(\mathbf u )+\overline{H}(\mathbf u )-1. \end{aligned} \end{aligned}$$

Thus,

$$\begin{aligned} \begin{aligned} \beta _n(C)&=\dfrac{2^{n-1}\left[ C\left( \frac{1}{2}{} \mathbf 1 _n\right) +\overline{C}\left( \frac{1}{2}{} \mathbf 1 _n\right) \right] -1}{2^{n-1}-1}\\&=\dfrac{2^{n-1}\left[ C^*\left( \frac{1}{2}{} \mathbf 1 _n\right) +H\left( \frac{1}{2}{} \mathbf 1 _n\right) +\overline{C^*}\left( \frac{1}{2}{} \mathbf 1 _n\right) +\overline{H}\left( \frac{1}{2}{} \mathbf 1 _n\right) -1\right] -1}{2^{n-1}-1}\\&=\dfrac{2^{n-1}\left[ C^*\left( \frac{1}{2}{} \mathbf 1 _n\right) +\overline{C^*}\left( \frac{1}{2}{} \mathbf 1 _n\right) \right] -1 +2^{n-1}\left[ H\left( \frac{1}{2}{} \mathbf 1 _n\right) +\overline{H}\left( \frac{1}{2}{} \mathbf 1 _n\right) \right] -1+1-2^{n-1}}{2^{n-1}-1}\\&=\beta _n(C^*)+\beta _n(H)+\dfrac{1-2^{n-1}}{2^n-1}. \end{aligned} \end{aligned}$$

      $\square$

Remark 2

In the above theorem, although the perturbation function H is not a copula, we still use m(H) to denote the corresponding values of H, where $m=\tau _n$, $\rho _n,$ or $\beta _n$, and use $\overline{H}$ to denoted the corresponding function of H defined by (1). The similar notations are used in the following context.

Remark 3

As n increasing, we can see that

$$\tau _n(C)\approx \tau _n(C^*)+\tau _n(H)+2\int _{I^n}C^*(\mathbf u )dH(\mathbf u )+2\int _{I^n}H(\mathbf u )dC^*(\mathbf u ),$$

$$\rho _n(C)\approx \rho _n(C^*)+\rho _n(H),$$

and

$$\beta _n(C)\approx \beta _n(C^*)+\beta _n(H).$$

Corollary 2

Let C be an n-copula defined by (7), then the Kendall’s tau, Spearman’s rho, and Blomqvist’s beta of C are given by

$$\begin{aligned} \tau _n(C)=\tau _n(C^*)+\tau _n(\theta \overset{n}{\underset{i=1}{\prod }}f_i)+a_2, \end{aligned}$$

(14)

$$\begin{aligned} \rho _n(C)=\rho _n(C^*)+\rho _n(\theta \overset{n}{\underset{i=1}{\prod }}f_i)+\dfrac{n+1}{2^{n}-n-1}, \end{aligned}$$

(15)

$$\begin{aligned} \beta _n(C)=\beta _n(C^*)+\beta _n(\theta \overset{n}{\underset{i=1}{\prod }}f_i)+\dfrac{1-2^{n-1}}{2^n-1}, \end{aligned}$$

(16)

where $a_2=\dfrac{1}{2^{n-1}-1}\left[ 2^n\int _{I^n}\theta C^*(\mathbf u )\overset{n}{\underset{i=1}{\prod }}{f_i}^\prime (u_i)d\mathbf u +2^n\int _{I^n}\theta \overset{n}{\underset{i=1}{\prod }}f_i(u_i)dC^*(\mathbf u )+1\right]$.

In 2013, Tasena et al. [17] defined a measure of multivariate complete dependence as follows. Let C be an n-copula of random variables $X_1, \cdots , X_n$. Define

$$\delta _i(X_1, \cdots , X_n)=\delta _i(C)=\dfrac{\int (\partial _iC-\pi _iC)^2}{\int \pi _iC(1-\pi _iC)}, $$

where $\pi _iC: I^{n-1}\rightarrow I$ is defined by

$$\pi _iC(u_1, \cdots , u_{n-1}) =C(u_1, \cdots , u_{i-1}, 1, u_{i}, \cdots , u_{n-1}),\qquad i=1, 2, \cdots , n.$$

By Theorem 3.6 in [17], $\delta _i$ satisfies following properties,

• (i) $0\le \delta _i(C)\le 1$,

• (ii) $\delta _i(C)=1$ if and only if $(X_1, \cdots , X_{i-1}, X_{i+1}, \cdots , X_{n})$ is a function of $X_i$. For details, see [17].

Theorem 6

Let C be an n-copula defined by (5). If

$$H(u_1, \cdots , u_{i-1}, 1, u_{i+1}, \cdots , u_{n_1})=0,$$

then

$$\delta _i(C)=\delta (C^*)-\dfrac{\int 2\partial _iH(\partial _iC^*-\pi _iC^*)-(\partial _iH)^2}{\int \pi _iC^*(1-\pi _iC^*)}.$$

Proof

By the definition,

$$\begin{aligned} \begin{aligned} \delta _i(C)&=\dfrac{\int (\partial _iC-\pi _iC)^2}{\int \pi _iC(1-\pi _iC)}\\&=\dfrac{\int (\partial _i(C^*+H)-\pi _i(C^*+H))^2}{\int \pi _i(C^*+H)[1-\pi _i(C^*+H)]}\\&=\dfrac{\int (\partial _iC^*+\partial _iH-\pi _iC^*-\pi _iH)^2}{\int (\pi _iC^*+\pi _iH)(1-\pi _iC^*-\pi _iH)}. \end{aligned} \end{aligned}$$

If $\pi _iH(u_1, \cdots , u_{n_1})=H(u_1, \cdots , u_{i-1}, 1, u_{i+1}, \cdots , u_{n_1})=0,$ then

$$\begin{aligned} \begin{aligned} \delta _i(C)&=\dfrac{\int (\partial _iC^*+\partial _iH-\pi _iC^*-\pi _iH)^2}{\int (\pi _iC^*+\pi _iH)(1-\pi _iC^*\pi _iH)}\\&=\dfrac{\int (\partial _iC^*+\partial _iH-\pi _iC^*)^2}{\int \pi _iC^*(1-\pi _iC^*)} \\&=\dfrac{\int \left[ (\partial _iC^*-\pi _iC^*)^2-2\partial _iH(\partial _iC^*-\pi _iC^*)+(\partial _iH)^2\right] }{\int \pi _iC^*(1-\pi _iC^*)}\\&=\dfrac{\int (\partial _iC^*-\pi _iC^*)^2}{\int \pi _iC^*(1-\pi _iC^*)}-\dfrac{\int 2\partial _iH(\partial _iC^*-\pi _iC^*)-(\partial _iH)^2}{\int \pi _iC^*(1-\pi _iC^*)}\\&=\delta (C^*)-\dfrac{\int 2\partial _iH(\partial _iC^*-\pi _iC^*)-(\partial _iH)^2}{\int \pi _iC^*(1-\pi _iC^*)}. \end{aligned} \end{aligned}$$

      $\square$

Corollary 3

Let C be an n-copula defined by (7). If $f_i(1)=0,$ then

$$ \delta _i(C)=\delta _i(C^*)-\dfrac{\int 2\theta f_i^{\prime } \prod \limits _{j\ne i} f_j(\partial _iC^*-\pi _iC^*)-(\theta f_i^{\prime } \prod \limits _{j\ne i} f_j)^2}{\int \pi _iC^*(1-\pi _iC^*)}. $$

Now, let’s recall some dependence concepts of copulas. For details, see [6, 11]. Let $C_1$ and $C_2$ be two n-copulas. If $C_1\ge C_2$ ($\overline{C}_1\ge \overline{C}_2$ resp.), i.e., $C_1(\mathbf u )\ge C_2(\mathbf u )$ ($\overline{C}_1(\mathbf u )\ge \overline{C}_2(\mathbf u )$ resp.) for all $\mathbf u \in I^n$, then we say that $C_1$ is more positive lower (upper resp.) orthant dependent (PLOD) (PUOD resp.) than $C_2$. $C_1$ is more positive orthant dependent (POD) than $C_2$ if $C_1\ge C_2$ and $\overline{C}_1\ge \overline{C}_2$ hold.

The following results give us some dependence relations between C and $C^*$. The proof is trivial.

Proposition 1

Let $C_1$ and $C_2$ be two n-copulas defined by (5). If they share the same n-copula $C^*$ and may have different perturbation functions $H_i$ $i=1,2$, then

(i) $C_1$ more PLOD than $C_2$ if and only if $H_1\ge H_2$,

(ii) $C_1$ more PUOD than $C_2$ if and only if $\overline{H}_1\ge \overline{H}_2$,

(iii) $C_1$ more POD than $C_2$ if and only if $H_1\ge H_2$ and $\overline{H}_1\ge \overline{H}_2$.

Proposition 2

Let $C_1$ and $C_2$ are two n-copulas defined by (7). If they share the same known n-copula $C^*$ and may have different perturbation functions $f_{j1}, \cdots , f_{jn}$, and parameters $\theta _j$, $j=1,2$ respectively, then

(i) $C_1$ more PLOD than $C_2$ if and only if $\theta _1\overset{n}{\underset{i=1}{\prod }}f_{1i}\ge \theta _2\overset{n}{\underset{i=1}{\prod }}f_{2i}$,

(ii) $C_1$ more PUOD than $C_2$ if and only if $\overline{\theta _1\overset{n}{\underset{i=1}{\prod }}f_{1i}}\ge \overline{\theta _2\overset{n}{\underset{i=1}{\prod }}f_{2i}}$,

(iii) $C_1$ more POD than $C_2$ if and only if $\theta _1\overset{n}{\underset{i=1}{\prod }}f_{1i}\ge \theta _2\overset{n}{\underset{i=1}{\prod }}f_{2i}$ and $\overline{\theta _1\overset{n}{\underset{i=1}{\prod }}f_{1i}}\ge \overline{\theta _2\overset{n}{\underset{i=1}{\prod }}f_{2i}}$.

The next theorem give us a property of the construction (6).

Theorem 7

Let $(U_1^*, \cdots , U_n^*)$ and $(U_1, \cdots , U_n)$ be random vectors with uniform marginals on [0, 1] and connected by copulas $C^*$ and C respectively. C and $C^*$ satisfy conditions of Theorem 3. Suppose that $f_i(1)=f_j(1)=0$, $1\le i<j\le n$.

(i) If there is $1\le l \le n$ such that $l\ne i, j$ and $f_l(1)=0$, then $P\{U_i<U_j\}=P\{U_i^*<U_j^*\}$,

(ii) If $f_l(1)\ne 0$ for all $l\ne i, j$ and $f_i=f_j$, then $P\{U_i<U_j\}=P\{U_i^*<U_j^*\}$.

Proof

(i) Let c and $c^*$ be the densities of C and $C^*$ respectively, then we have

$$c(\mathbf u )=\dfrac{\partial ^nC(\mathbf u )}{\partial u_1\cdots \partial u_n}=c^*(\mathbf u )+ \overset{n}{\underset{i=1}{\prod }}{f_i}^\prime (u_i).$$

Then

$$\begin{aligned} \begin{aligned} P\{U_i<U_j\}&=\int _0^1\cdots \int _0^{u_j}\cdots \int _0^1c(u_1,\cdots , u_i,\cdots , u_n)du_1\cdots du_i \cdots du_n \\&=\int _0^1\cdots \int _0^{u_j}\cdots \int _0^1c^*(u_1,\cdots , u_i,\cdots , u_n)du_1\cdots du_i \cdots du_n\\&\ \ \ \ +\int _0^1\cdots \int _0^{u_j}\cdots \int _0^1 f^\prime _1(u_i)\cdots f^\prime _i(u_i)\cdots f^\prime _n(u_n) du_1\cdots du_i\cdots du_n\\&=P\{U^*_i<U^*_j\}+ \overset{n}{\underset{k\ne i, j}{\prod }}(f_k(1)-f_k(0))\int _0^1\int _0^{u_j}f^\prime _j(u_j)f^\prime _i(u_i)du_idu_j\\&=P\{U^*_i<U^*_j\}, \end{aligned} \end{aligned}$$

since $f_l(0)=f_l(1)=0$ and $l\ne i, j$.

(ii) Similarly, if $f_l(1)\ne 0$ for all $l\ne i, j$,

$$\begin{aligned} \begin{aligned} P\{U_i<U_j\}&=P\{U^*_i<U^*_j\}+ \overset{n}{\underset{k\ne i, j}{\prod }}(f_k(1)-f_k(0))\int _0^1\int _0^{u_j}f^\prime _j(u_j)f^\prime _i(u_i)du_idu_j\\&=P\{U^*_i<U^*_j\}+ \overset{n}{\underset{k\ne i, j}{\prod }}f_k(1)\int _0^1\int _0^{u_j}f^\prime _j(u_j)f^\prime _i(u_i)du_idu_j\\&=P\{U^*_i<U^*_j\}+ \overset{n}{\underset{k\ne i, j}{\prod }}f_k(1)\int _0^1f_i(u_j)f^\prime _j(u_j)du_j. \end{aligned} \end{aligned}$$

Since $f_i=f_j$,

$$\begin{aligned} \begin{aligned} \int _0^1f_i(u_j)f^\prime _j(u_j)du_j&=f_j(1)f_i(1)-f_j(0)f_i(0)-\int _0^1f_j(u_j)f^\prime _i(u_j)du_j\\&=-\int _0^1f_j(u_j)f^\prime _i(u_j)du_j=-\int _0^1f_i(u_j)f^\prime _j(u_j)du_j, \end{aligned} \end{aligned}$$

and hence $\int _0^1f_i(u_j)f^\prime _j(u_j)du_j=0$. So $P\{U_i<U_j\}=P\{U_i^*<U_j^*\}.$       $\square$

The following example shows that the converse of the above result (ii) in Theorem 7 may not hold in general. Moreover, it shows that Theorem 3 in [7] is incorrect.

Example 1

Let $(U^*, V^*)$ and (U, V) be random vectors with uniform marginals on [0, 1]. Suppose that $(U^*, V^*)$ is connected by the independent copula, i.e., $C^*(u, v)=uv$, and (U, V) is connected by $C(u, v)=C^*(u, v)+f(u)g(v)$, where $f(u)=u(1-u)$, $g(v)=\dfrac{1}{2}v(1-v)$. Then f and g satisfy the conditions in Theorem 3. In fact, C belongs to the bivariate FGM family.

As the proof of the above theorem, we have

$$\begin{aligned} \begin{aligned} P\{U<V\}&=\int _0^1\int _0^v c(u, v)dudv =\int _0^1\int _0^v c^*(u, v)+f^\prime (u)g^\prime (v)dudv \\&=P\{U^*<V^*\}+\int _0^1\int _0^v f^\prime (u)g^\prime (v)dudv=P\{U^*<V^*\}+\int _0^1 f(v)g^\prime (v)dv.\\ \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} \int _0^1 f(v)g^\prime (v)dv&= \int _0^1 \frac{1}{2}v(1-v)(1-2v)dv= 0. \end{aligned} \end{aligned}$$

Thus $P\{U<V\}=P\{U^*<V^*\}$, but $f\ne g$.

5 Examples

In this section, we provide two examples. The given copula $C^*$ in the first example is the simplest one, the independent copula. To emphasis multivariate and for simplicity, we will only consider 3-copulas, but results could be extended to n-copulas. In the second example, $C^*$ is nontrivial. Also for simplicity, we will only consider 2-copulas.

Example 2

Let $C^*$ be the independent 3-copula, i.e., $C^*(u,v,w)=uvw$. Let $f(x)=x(1-x^k)$, where $u, v, w, x \in I$, $k\in \mathbb {N}$, the set of all positive integers. Consider the 3-copula family,

$$\begin{aligned} \begin{aligned} C(u, v, w)&= C^*(u,v,w)+\theta f(u)f(v)f(w)\\&= uvw+\theta uvw(1-u^k)(1-v^k)(1-w^k), \end{aligned} \end{aligned}$$

where $\theta \in \mathbb {R}$.

It is clear that f(x) satisfies the conditions (i) and (ii) of Corollary 1. Next we will use the condition (iii) of Corollary 1 to find the range of the parameter $\theta$ for each k. Firstly, it is easy to see that $\dfrac{V_{C^*}([\mathbf u , \mathbf v ])}{\varDelta (\mathbf u , \mathbf v )}=1$ for any $\mathbf u , \mathbf v \in [0, 1]^3$ with $\mathbf u <\mathbf v$. Secondly, $f^\prime (x)=1-(k+1)x^k$, so

$$\alpha =\inf \{f^\prime (x): x\in I\}=f^\prime (1)=1-(k+1)=-k,$$

and

$$\beta =\sup \{f^\prime (x): x\in I\}=f^\prime (0)=1.$$

Thus, as the notations in Theorem 3, $B=\{-k, -k^3\}$, $B^\prime =\{k^2\}$. So by the condition (iii) of Corollary 1, the range of $\theta$ is

$$-\dfrac{1}{\max (B^\prime )}\le \theta \le -\dfrac{1}{\min (B)}, $$

i.e.,

$$-\dfrac{1}{k^2}\le \theta \le \dfrac{1}{k^3}.$$

So, we can see that the range of $\theta$ is shrinking as k increasing. Specifically, if $k=1$, $-1\le \theta \le 1$. If $k=2$, $-\dfrac{1}{4}\le \theta \le \dfrac{1}{8}.$ If $k=3$, $-\dfrac{1}{9}\le \theta \le \dfrac{1}{27}.$

Next, let’s compute three measures discussed in Sect. 4 for these 3-copulas. By the definition of $\tau _n$,

$$\tau _3\left( \theta f(u)f(v)f(w)\right) =\dfrac{1}{3}\left[ 8\theta (\dfrac{k+1}{2k+3}-\dfrac{k+2}{k+3}+\dfrac{1}{3})-1\right] .$$

$$\begin{aligned} \begin{aligned} a_2&=\dfrac{1}{3}\left[ 8\int _{I^3}\theta C^*(\mathbf u )\overset{3}{\underset{i=1}{\prod }}{f_i}^\prime (u_i)d\mathbf u +8\int _{I^3}\theta \overset{3}{\underset{i=1}{\prod }}f_i(u_i)dC^*(\mathbf u )+1\right] \\&=\dfrac{1}{3}\left[ -\dfrac{\theta k^3}{(k+2)^3}+\dfrac{\theta k^3}{(k+2)^3}+1\right] = \dfrac{1}{3}. \end{aligned} \end{aligned}$$

So by Corollary 2,

$$\begin{aligned} \begin{aligned} \tau _3(C)&=\tau _3(C^*)+\tau _3(\theta f(u)f(v)f(w))+a_2\\&= 0+\dfrac{1}{3}\left[ 8\theta \left( \dfrac{k+1}{2k+3}-\dfrac{k+2}{k+3}+\dfrac{1}{3}\right) -1\right] +\dfrac{1}{3}\\&= \dfrac{8\theta }{3}\left( \dfrac{k+1}{2k+3}-\dfrac{k+2}{k+3}+\dfrac{1}{3}\right) . \end{aligned} \end{aligned}$$

So the range of $\tau _3(C)$ is

$$\dfrac{8}{3k^3}\left( \dfrac{k+1}{2k+3}-\dfrac{k+2}{k+3}+\dfrac{1}{3}\right) \le \tau _3(C)\le -\dfrac{8}{3k^2}\left( \dfrac{k+1}{2k+3}-\dfrac{k+2}{k+3}+\dfrac{1}{3}\right) .$$

By the definition of $\rho _n$,

$$\rho _3(\theta f(u)f(v)f(w))=4\left[ \dfrac{\theta k^3}{8(k+2)^3}-\dfrac{\theta k^3}{8(k+2)^3}\right] -1=-1.$$

So

$$\begin{aligned} \begin{aligned} \rho _3(C)&=\rho _3(C^*)+\rho _3(\theta f(u)f(v)f(w))+\dfrac{3+1}{2^3-3-1}\\&= 0-1+1 = 0. \end{aligned} \end{aligned}$$

By the definition of survival function (1),

$$\overline{\theta \overset{3}{\underset{i=1}{\prod }}f_i}\left( \frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right) =1-\dfrac{\theta }{8}\left( 1-\dfrac{1}{2^k}\right) ^3.$$

So

$$\begin{aligned} \begin{aligned} \beta _3(\theta f(u)f(v)f(w))&=\dfrac{2^{2}\left[ \theta \overset{3}{\underset{i=1}{\prod }}f_i\left( \frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right) +\overline{\theta \overset{3}{\underset{i=1}{\prod }}f_i}\left( \frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right) \right] -1}{2^{2}-1}\\&=\dfrac{4\left[ \dfrac{\theta }{8}\left( 1-\dfrac{1}{2^k}\right) ^3 +1-\dfrac{\theta }{8}\left( 1-\dfrac{1}{2^k}\right) ^3\right] -1}{3}=1. \end{aligned} \end{aligned}$$

Thus,

$$\begin{aligned} \begin{aligned} \beta _3(C)&=\beta _3(C^*)+\beta _3(\theta f(u)f(v)f(w))+\dfrac{1-2^{3-1}}{2^3-1}\\&=0+1-\frac{3}{7}= \frac{4}{7}. \end{aligned} \end{aligned}$$

Lastly, since $f(u)f(v)f(w)=uvw(1-u^k)(1-v^k)(1-w^k)\ge 0$ for all $(u,v,w)\in I^3$, we have that C is more PLOD than $\varPi _3$ if and only if $\theta \ge 0$ and $\varPi _3$ is more PLOD than C if and only if $\theta \le 0$.

Remark 4

From the above example, we can see that this 3-copulas family, $C(u, v, w)= C^*(u, v, w)+\theta uvw(1-u^k)(1-v^k)(1-w^k)$, is interesting. As long as this C is a 3-copula, $\rho _3(C)$ and $\beta _3(C)$ are free of $\theta$. Specifically, we always have $\rho _3(C)=\rho _3(C^*)$ and $\beta _3(C)=\beta _3(C^*)+\frac{4}{7}$.

Example 3

Let $C^*$ be a Frank’s copula [2, 5] defined by

$$C^*(u, v)=\ln \left[ 1+\dfrac{(e^u-1)(e^v-1)}{e-1}\right] .$$

Let

$$H=\theta (1-u)(1-e^u)(1-v)(1-e^v),$$

where $\theta \ge 0$. Define a bivariate function C by $C=C^*+H.$ We will use Theorem 2 to find the range of $\theta$ such that C is a copula.

Firstly, it is easy to see that $H(0, v)=H(u, 0)=H(1, v)=H(u, 1)=0$.

Secondly, we can find that

$$c^*(u, v)=\dfrac{(e-1)(u+v)}{[e-1+(e^u-1)(e^v-1)]^2},$$

and

$$h(u, v)=\theta (ue^u-1)(ve^v-1).$$

It can be shown that minimum values of $c=c^*+h$ occur at (0, 1) and (1, 0). So $c\ge 0$ if and only if $c(0,1)=c(1,0)=c^*(0, 1)+h(0,1)=\dfrac{1}{e-1}-\theta (e-1)\ge 0$. Thus $C=C^*+H$ is a copula if $\theta \le \dfrac{1}{(e-1)^2}$.