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Unformatted text preview: What are copulas? Jos e Juan Quesada-Molina Departamento de Matem atica Aplicada, Universidad de Granada Jos e Antonio Rodr guez-Lallena Manuel Ubeda-Flores Departamento de Estad stica y Matem atica Aplicada, Universidad de Almer a Monograf as del Semin. Matem. Garc a de Galdeano. 27: 499506, (2003). Abstract The notion of copula was introduced by A. Sklar in 1959, when answering a question raised by M. Fr echet about the relationship between a multidimensional probability function and its lower dimensional margins. At the beginning, copulas were mainly used in the development of the theory of probabilistic metric spaces. Later, they were of interest to define nonparametric measures of dependence between random variables, and since then, they began to play an important role in probability and mathematical statistics. In this paper, a general overview of the theory of copulas will be presented. Some of the main results of this theory, various examples, and some open problems will be described. MSC : Primary 60E05; Secondary 62H05, 62H20. Keywords : Copulas; dependence concepts; measures of association, probabilistic metric spaces. 1 Introduction During a long time statisticians have been interested on the relationship between a mul- tivariate distribution function and its lower dimensional margins (univariate or of higher dimensions). M. Fr echet (see ), and G. DallAglio (see ) did some interesting works about this matter in the fifties, studying the bivariate and trivariate distribution functions with given univariate margins. The answer to this problem for the univariate margins case was given by A. Sklar in 1959 (see ) creating a new class of functions which he called copulas . These new functions are restrictions to [0 , 1] 2 of bivariate distribution functions 499 whose margins are uniform in [0 , 1]. In short, Sklar showed that if H is a bivariate dis- tribution function with margins F ( x ) and G ( y ), then there exists a copula C such that H ( x, y ) = C ( F ( x ) , G ( y )) . Between 1959 and 1976 most of the results about copulas were obtained in the course of the development of the probabilistic metric spaces, mainly in the study of binary operations in the space of the probability distribution functions. In 1942, Karl Menger (see ) proposed a probabilistic generalization of the theory of metric spaces, by replacing the number d ( p, q ) by a distribution function F pq , whose value F pq ( x ) for any real x is the probability that the distance between p and q is less than x . The first difficulty in the construction of probabilistic metric spaces comes when one tries to find a probabilistic analog of the triangle inequality. Menger proposed F pr ( x + y ) T ( F pq ( x ) , F qr ( y )), where T is a triangle norm or t-norm . Some t-norms are copulas, and conversely, some copulas are t-norms. For a history of the development of the theory of probabilistic metric spaces, see  and .see  and ....
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