Multivariate Copula Expressed by Lower Dimensional Copulas

TL;DR

This paper introduces a theorem that expresses high-dimensional multivariate copulas using lower-dimensional copulas based on conditional independence, addressing the curse of dimensionality in modeling complex distributions.

Contribution

It presents a new theorem for representing multivariate copulas via lower-dimensional ones and introduces the sample derivated copula to facilitate this construction.

Findings

Theorem enables expressing multivariate copulas with lower-dimensional copulas.

Sample derivated copula depends only on the copula and partition.

Application to construct multivariate discrete copulas from marginals.

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.