Why Copulas Have Been Successful in Many Practical Applications: A Theoretical Explanation Based on Computational Efficiency

Abstract

A natural way to represent a 1-D probability distribution is to store its cumulative distribution function (cdf) $F(x)=\mathrm{Prob}(X\le x)$. When several random variables $X_1,\ldots ,X_n$ are independent, the corresponding cdfs $F_1(x_1)$, ..., $F_n(x_n)$ 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 $F(x_1,\ldots ,x_n)=\mathrm{Prob}(X_1\le x_1\, \& \,\ldots \, \& \,X_n\le x_n)$, 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 $F(x_1,\ldots ,x_n)$. In this paper, we prove that among all duplication-free representations, the most computationally efficient one 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: since the copula representation is, in some reasonable sense, the most computationally efficient way of representing multi-D probability distributions.