Properties and applications of copulas: A brief survey

A copula is a function which joins or “couples” a multivariate distribution function to its one-dimensional marginal distribution functions. The word “copula” was first used in a mathematical or statistical sense by Sklar (1959) in the theorem which bears his name (see the next section). But the functions themselves predate the use of the term, appearing in the work of Hoeffding, Fréchet, Dall’Aglio, and many others. Over the past forty years or so, copulas have played an important role in several areas of statistics. As Fisher (1997) notes in the Encyclopedia of Statistical Sciences, “Copulas [are] of interest to statisticians for two main reasons: First, as a way of studying scale-free measures of dependence; and secondly, as a starting point for constructing families of bivariate distributions, ...” In Sections 2 through 5 we present the basic properties of copulas and several families of copulas useful in statistical modeling, and in Sections 6 and 7 we explore the relationships between copulas and dependence properties and measures of association. The concept of a quasi-copula was introduced by Alsina et al. (1993) in order to characterize operations on distribution functions that can or cannot be derived from operations on random variables. We discuss quasicopulas and their relationship with copulas in Sections 8 and 9. We conclude with extensions to higher dimensions in Section 10, and a few open problems. This brief survey is necessarily incomplete. Readers seeking to learn more about copulas and quasi-copulas will find the monographs by Hutchinson and Lai (1990), Joe (1997), and Nelsen (1999) and conference proceedings edited by Beneš and Štěpán (1997), Cuadras et al. (2002), Dall’Aglio et al. (1991), and Rüschendorf et al. (1996) useful.