Asian Journal of Mathematics & Statistics Year: 2009 | Volume: 2 | Issue: 3 | Page No.: 41-54 DOI: 10.3923/ajms.2009.41.54 URL: http://scialert.net/abstract/?doi=ajms.2009.41.54 Research Article Modelling the Dependence of Parametric Bivariate Extreme Value Copulas S. Dossou-Gbete , B. Some and D. Barro * * ABSTRACT In this study, we consider the situation where contraints are made on the domains of two random variables whose joint copula is an extreme value model. We introduce a new measure which characterize these conditional dependence. We proved that every bivariate extreme value copulas is totally characterized by a conditional dependence function. Every two- dimensional distribution is also shown to be max-infinite divisible under a restriction on the new measure. The average and median values of the measure have been computed for the main bivariate families of parametric extreme value copulas. INTRODUCTION Modelling the dependence between several random variables lies at the heart of the subject of Extreme Value Theory (EVT). In this theory, many structures have been developed to describe and to measure this dependence (Pickands, 1981; Coles, 2001). Some of these structures remain invariant under strictly increasing transformations of lower dimensional marginal variables. It is the case of copulas, multivariate distributions whose univariate margins are uniformly distributed on the unit interval I = [0, 1] and which establish, via Sklar theorem, a close connection between every multivariate distribution function and its univariate margins (Sklar, 1996; Joe, 1997). In Bivariate Extreme Value (BEV) study, the tail dependence parameters estimate numerically the importance of asymptotic dependence between two random variables. Let H be the distribution function of a random pair (X 1 , X 2 ) with univariate margins {H i , i = 1, 2}. The tail dependence parameter λ of H, given by: (1) Quantifies the probability to observe a large X 1 assuming that X 2 is also large, where H and denote, respectively the survivor function and the right endpoint of H. In many latest studies published on the topic of EVT and applications (Beirlant et al., 2005; Michel, 2006), it has been shown that no unique parametric structure can summary the family of multivariate distributions like in univariate situation. Nevertheless, if the univariate margins are given, the dependence of the joint distribution can be characterized by one of equivalent measures like Pickands dependence function (Pickands, 1981), exponent measure or stable tail dependence function (Michel, 2006; Degen, 2006). However, these measures do not take into account the conditions that would be made on lower dimensional margins. The main contribution of this study is to construct a new measure and function which describe * Corresponding author : Diakarya Barro, 03 BP: 7021 Ouagadougou 03, UFR-SEA, tel: (226) 70 23 66 13