Thirty Years of Copulas

Abstract

In 1959, in response to a query of M. Fréchet, A. Sklar introduced copulas. These are functions that link multivariate distributions to their one-dimensional margins. Thus, if H is an n-dimensional cumulative distribution function with one-dimensional margins F1,…,Fn, then there exists an n-dimensional copula C (which is unique when F1,…,Fn are continuous) such that H(x1,…,xn) = C (F1(x1),…,Fn (xn)). During the years 1959 — 1974, most results concerning copulas were obtained in the course of the development of the theory of probabilistic metric spaces, principally in connection with the study of families of binary operations on the space of probability distribution functions. Then it was discovered that two-dimensional copulas could be used to define nonparametric measures of dependence for pairs of random variables. In the ensuing years the copula concept was rediscovered on several occasions and these functions began to play an ever-more-important role in mathematical statistics, particularly in matters involving questions of dependence, fixed marginals and functions of random variables that are invariant under monotone transformations. Today, in view of the fact that they are the higher dimensional analogues of uniform distributions on the unit interval, and as the result of the efforts of a diverse group of scholars, the significance, ubiquity and utility of copulas is being recognized. This paper is devoted to an historical overview and rather personal account of these developments and to a description of some recent results.