Set-up

An $n$ -dimensional rectangle $S$ is a subset of $\mathbb{R}^n$ of the form $I_1\times \cdots \times I_n$ , where each $I_k$ is an interval, with end points $a_k\le b_k\in \mathbb{R}^*$ , where $\mathbb{R}^*$ is the set of extended real numbers (so that $\mathbb{R}$ itself may be considered as an interval).

Groundedness. A function $C:S\to \mathbb{R}$ is said to be grounded if for each $1\le k\le n$ , and each $r_j\in I_j$ where $j\ne k$ , the function $C_k:I_k\to \mathbb{R}$ defined by $$C_k(x):=C(r_1,\ldots,r_{j-1},x,r_{j+1},\ldots,r_n)$$ is right-continuous at $a_k$ , the lower end point of $I_k$ .

Margin. Note that $C_k$ defined above may or may not exist as each $r_j\to b_j$ , the upper end point of $I_j$ ($j\ne k$ ). If the limit exists, then we call this limiting function, also written $C_k$ , a (one-dimensional) margin of $C$ : $$C_k(x):=\lim_{r_j\to b_j}\ C(r_1,\ldots,r_{j-1},x,r_{j+1},\ldots,r_n),\mbox{ where }j\in\lbrace 1,\ldots,n\rbrace\mbox{, }j\neq i.$$

Given an $n$ -dimensional rectangle $S=I_1\times \cdots \times I_n$ , let's call each $I_k$ a side of $S$ . A vertex of $S$ is a point $v\in\mathbb{R}^n$ such that each of its coordinates is an end point. Clearly $S$ is a convex set and the sides and vertices lie on the boundary of $S$ .

$C$ -volume. Suppose we have a function $C:S\to \mathbb{R}$ , with $S$ defined as above. Let $T$ be a closed $n$ -dimensional rectangle in $S$ ($T\subseteq S$ ), with sides $J_k=[c_k,d_k]$ , $1\le k\le n$ . The $C$ -volume of $T$ is the sum $$\operatorname{Vol}_C(T)=\sum (-1)^{n(v)}C(v)$$ where $v$ is a vertex of $T$ , $n(v)$ is the number of lower end points that occur in the coordinate representation of $v$ , and the sum is taken over all vertices of $T$ .

The name volume is derived from the fact that if $C(x_1,\ldots,x_n)=x_1\cdots x_n$ , then for each closed rectangle $T$ , $\operatorname{Vol}_C(T)$ is the volume of $T$ in the traditional sense.

Note, however, depending on the function $C$ , $\operatorname{Vol}_C(T)$ may be $0$ or even negative. For example, if $C$ is a linear function, then the $C$ -volume is identically $0$ for every closed rectangle $T$ , whenever $n$ is even. An example where $\operatorname{Vol}_C(T)$ is negative is given by the function $C(x,y)=-xy$ , and $T$ is the unit square.

$n$ -increasing. A function $C:S\to\mathbb{R}$ where $S$ is an open $n$ -dimensional rectange is said to be $n$ -increasing if $\operatorname{Vol}_C$ is non-negative evaluated at each closed rectangle $T\subseteq S$ .

Any multivariate distribution function is both grounded and $n$ -increasing.

Definition

A copula, introduced by Sklar, is both a variant and a generalization of a multivariate distribution function.

Formally, a copula is a function $C$ from the $n$ -dimensional unit cube $I^n$ ($I=[0,1]$ ) to $\mathbb{R}$ satisfying the following conditions:

1. $C$ is $n$ -increasing,
2. $C$ is grounded,
3. every margin $C_k$ of $C$ is the identity function.

If we replace the domain by any $n$ -dimensional rectangle $S$ , then the resulting function is called a subcopula.

For example, the functions $C(x,y,z)=xyz$ , $C(x,y,z)=\min(x,y,z)$ , and $C(x,y,z)=\max(0,(x+y+z-2))$ defined on the unit cube are all copulas.

(This entry is in the process of being expanded, more to come shortly).

Bibliography

1

B. Schweizer and A. Sklar, Probabilistic Metric Spaces, Dover Publications, (2005).