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# copula

# 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 $a_{k}\leq b_{k}\in\mathbb{R}^{*}$, 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 $1\leq k\leq n$, and each $r_{j}\in I_{j}$ where $j\neq k$, 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 $I_{j}$ ($j\neq k$). If the limit exists, then we call this limiting function, also written $C_{k}$, a (one-dimensional) *margin* of $C$:

$C_{k}(x):=\lim_{{r_{j}\to b_{j}}}\ C(r_{1},\ldots,r_{{j-1}},x,r_{{j+1}},\ldots% ,r_{n}),\mbox{ where }j\in\{1,\ldots,n\}\mbox{, }j\neq i.$ |

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 $J_{k}=[c_{k},d_{k}]$, $1\leq k\leq n$. 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).

# References

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

## Mathematics Subject Classification

62A01*no label found*54E70

*no label found*

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