Multivariate Copula Expressed by Lower Dimensional Copulas

Abstract

Modeling of high order multivariate probability distribution is a difficult problem which occurs in many fields. Copula approach is a good choice for this purpose, but the curse of dimensionality still remains a problem. In this paper we give a theorem which expresses a multivariate copula by using only some lower dimensional ones based on the conditional independences between the variables. In general the construction of a multivariate copula using this theorem is quite difficult, due the consistency properties which have to be fulfilled. For this purpose we introduce the sample derivated copula, and prove that the dependence between the random variables involved depends just on this copula and on the partition. By using the sample derivated copula the theorem can be successfully applied, in order to to construct a multivariate discrete copula by using some of its marginals.