CopulasΒΆ
Let be a multivariate distribution function of dimension whose marginal distribution functions are . There exists a copula of dimension such that for , we have:
where is the cumulative distribution function of the margin .
In the case of continuous marginal distributions, for all , the copula is uniquely defined by:
where is a random variable following the uniform distribution on .
A copula of dimension is the restriction to the unit cube of a multivariate distribution function with uniform univariate marginals on . It has the following properties:
,
for all with at least one component equal to 0, ,
is -increasing which means that:
where and for all and , , ,
with all its components equal to 1 except , .
The copula represents the part of the joint cumulative density function which is not described by the marginal distributions. It models the dependence structure of the input variables.
Note that a multivariate distribution is characterized by its marginal distributions and its copula. Therefore, a multivariate distribution can be built by choosing the marginals and the copula independently.