Copulae and Their Uses

This short survey of copulas presents their properties, tries to stress their relevance for statistics and their connection with Markov processes and conditional expectations. 1 What is a copula? Copul were introduced by Sklar(1959). The reader is referred to Schweizer and Sklar (1983), Schweizer (1991), Sklar (1973), Sklar (1996), Nelsen (1999). A copula is a function C : [0; 1] ]0; 1[! ]0; 1[ that satis es the following properties: { for all t in [0; 1], C(t; 0) = C(0; t) = 0; { for all t in [0; 1], C(t; 1) = C(1; t) = t; { if x; x; y; y are in [0; 1] with x x and y y, then C(x; y) C(x; y) C(x; y) + C(x; y) 0: As a consequence of these properties it follows that (a) C satis es the Lipschitz condition jC(x; y) C(x; y)j jx xj+ jy yj; (b) that it is non{decreasing in each variable and (c) that it is absolutely continuous. In other words, a copula is a two{dimensional distribution function that concentrates all the probability mass on the unit square [0; 1] [0; 1] and which has uniform marginals. The importance of the concept of copula stems from the following Theorem 1. (Sklar). Let X and Y two random variables on the probability space ( ;F ; P ) having H as their joint distribution function and let F and G be the marginals of H, F (x) = H(x;+1); G(y) = H(+1; y): Then there exists (at least) a copula C such that H(x; y) = C (F (x); G(y)) : (1) If both F and G are continuous, then the copula C is uniquely determined. If either F or G, or both, is not continuous, then there may be more than one copula may satisfy (1); all of these coincide on the set RanF RanG. By a method of bilinear interpolation, which will always be adopted in the sequel, it is possible to choose a single copula that satis es (1). For this method see Lemma 2.3.5 in Nelsen (1999). The partial derivatives D1 C(x; y) := @C(x; y) @x ; D2 C(x; y) := @C(x; y) @y exist almost everywhere and are almost everywhere non{decreasing. 2 Statistical Proterties Copul are widely used in non{parametric statistics, especially in the study of dependence of random variables and in order to express the known measures of association between random variables, Kendall's tau, Spearman's rho, Gini's coeÆcient, see Nelsen (1999) and the bibliography quoted therein. A new measure of dependence was introduced by Schweizer and Wol (1981) in terms of the copula of the two random variables involved. Recently the work of some authors (Averous, Bassan, Dortet{Bernadet, and Spizzichino) has introduced the use of copulas in the study of multivariate aging. In a few cases, the multivariate aging function is not a copula but a quasi{copula (for which see Genet et al. (1999)). 3 Copul and Markov Processes The connection between Copulas and Markov processes is established through an operation on the set C of all copulas, which was introduced by Darsow, Nguyen and Olsen (1992). Let A and B be copul ; if x and y are in [0; 1] an operation on C is de ned via