copula

Set-up

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

Groundedness. A function $C:S\to ℝ$ 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 $C_k:I_k\to ℝ$ defined by

$$C_k(x):=C(r_1,\mathrm{\dots },r_{j-1},x,r_{j+1},\mathrm{\dots },r_n)$$

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 $C$:

$$C_k(x):=\underset{r_j\to b_j}{lim}C(r_1,\mathrm{\dots },r_{j-1},x,r_{j+1},\mathrm{\dots },r_n),\text{ where }j\in \{1,\mathrm{\dots },n\}\text{, }j\ne i.$$

Given an $n$-dimensional rectangle $S=I_1\times \mathrm{\cdots }\times I_n$, let’s call each $I_k$ a side of $S$. A vertex of $S$ is a point $v\in {ℝ}^n$ 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 $C:S\to ℝ$, 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

$${\mathrm{Vol}}_C(T)=\sum {(-1)}^{n(v)}C(v)$$

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

Note, however, depending on the function $C$, ${\mathrm{Vol}}_C(T)$ 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 ${\mathrm{Vol}}_C(T)$ is negative is given by the function $C(x,y)=-xy$, and $T$ is the unit square.

$n$-increasing. A function $C:S\to ℝ$ where $S$ is an open $n$-dimensional rectange is said to be $n$-increasing if ${\mathrm{Vol}}_C$ 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 $ℝ$ 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).

Title	copula
Canonical name	Copula
Date of creation	2013-03-22 16:33:43
Last modified on	2013-03-22 16:33:43
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	11
Author	CWoo (3771)
Entry type	Definition
Classification	msc 62A01
Classification	msc 54E70
Related topic	MultivariateDistributionFunction
Related topic	ThinSquare
Defines	subcopula
Defines	$n$-increasing
Defines	grounded
Defines	margin