Why Copulas?

Authors

Vladik Kreinovich, The University of Texas at El PasoFollow
Hung T. Nguyen, New Mexico State University - Main CampusFollow
Songsak Sriboonchitta, Chiang Mai UniversityFollow
Olga Kosheleva, The University of Texas at El PasoFollow

Publication Date

5-2015

Comments

Technical Report: UTEP-CS-15-39

Abstract

A natural way to represent a 1-D probability distribution is to store its cumulative distribution function (cdf) F(x) = Prob(X ≤ x). When several random variables X₁, ..., X_n are independent, the corresponding cdfs F₁(x₁), ..., F_n(x_n) provide a complete description of their joint distribution. In practice, there is usually some dependence between the variables, so, in addition to the marginals F_i(x_i), we also need to provide an additional information about the joint distribution of the given variables. It is possible to represent this joint distribution by a multi-D cdf F(x₁, ..., x_n) = Prob(X₁ ≤ x₁ & ... & X_n ≤ x_n), but this will lead to duplication -- since marginals can be reconstructed from the joint cdf -- and duplication is a waste of computer space. It is therefore desirable to come up with a duplication-free representation which would still allow us to easily reconstruct F(x₁, ..., x_n). In this paper, we prove that the only such representation is a representation in which marginals are supplements by a copula. This result explains why copulas have been successfully used in many applications of statistics.