## Quick Reference

A function that relates a joint cumulative distribution function to the distribution functions of the individual variables. If the individual distribution functions are known, but the joint distribution is unknown, then a copula can be used to suggest a suitable form for the joint distribution.

Let F be the multivariate distribution function for the random variables *X*_{1}, *X*_{2},…, *X** _{n}* and let the cumulative distribution function of

*X*

*be F*

_{j}*(for all*

_{j}*j*). Define random variables

*U*

_{1},

*U*

_{2},…,

*U*

*by*

_{n}*U*

*=F*

_{j}*(*

_{j}*X*

*) for each*

_{j}*j*, so that the marginal distribution of each

*U*

*has a continuous uniform distribution in the interval (0, 1). Assume that for each value*

_{j}*u*

*there is a unique value*

_{j}*x*

*=F*

_{j}^{−1}(

*u*

*) and let the joint cumulative distribution function of*

_{j}*U*

_{1},

*U*

_{2},…,

*U*

*be C. Then , for all*

_{n}*u*

_{1},

*u*

_{2},…,

*u*

*in (0, 1), since*

_{n}*U*

*<*

_{j}*u*

*if and only if*

_{j}*X*

*< F*

_{j}_{j}

^{-1}(

*u*

*). The function C is called the copula. An equivalent equation to the above is C{F*

_{j}_{1}(

*x*

_{1}), F

_{2}(

*x*

_{2}),…, F

*(*

_{n}*x*

*)}=F(*

_{n}*x*

_{1},

*x*

_{2},…,

*x*

*), for all*

_{n}*x*

_{1},

*x*

_{2},…,

*x*

*, where*

_{n}*u*

*=F*

_{j}*(*

_{j}*x*

*) for each*

_{j}*j*. Sklar's theorem, formulated by Abe Sklar of the Illinois Institute of Technology and published in 1959, states that, for a given F, there is a unique C such that this equation holds.

Note that it may well be that it is not possible to express the inverse functions F_{j}^{-1} in a simple form (an example is the multivariate normal distribution).

Assuming that the copula and the marginal distribution functions are differentiable, the corresponding result for probability density functions is that

f(*x*_{1}, *x*_{2},…, *x** _{n}*)=c{F

_{1}(

*x*

_{1}), F

_{2}(

*x*

_{2}),…, F

*(*

_{n}*x*

*)} f*

_{n}_{1}(

*x*

_{1})f

_{2}(

*x*

_{2})…f

*(*

_{n}*x*

*).*

_{n}The trivial case where c {F_{1}(*x*_{1}), F(*x*_{2}),…, F(x* _{n}*)}=1 corresponds to the case where the

*n*

*X*-variables are independent. Thus the copula encapsulates the interdependencies between the

*X*-variables and is therefore also known as the dependence function. If c(

*u*

_{1},

*u*

_{2},…,

*u*

*) is the joint probability density function of*

_{n}*U*

_{1},

*U*

_{2},…,

*U*

*, then*

_{n}c(*u*_{1}, *u*_{2},…, *u** _{n}*)=f(

*x*

_{1},

*x*

_{2},…,

*x*

*)/{f*

_{n}_{1}(

*x*

_{1})f

_{2}(

*x*

_{2})…

*f*

*(*

_{n}*x*

*)},*

_{n}where , for each *j*.

**From:**
copula
in
A Dictionary of Statistics »

*Subjects:*
Probability and Statistics.