Propriétés statistiques des copules de valeurs extrêmes bidimensionnelles

Let (X,Y) be a bivariate random vector whose distribution function H(x,y) belongs to the class of bivariate extreme-value distributions. If F₁ and F₂ are the marginals of X and Y, then H(x,y) = C{F₁(x),F₂(y)}, where C is a bivariate extreme-value dependence function. This paper gives the joint distribution of the random variables Z = {log F₁(X)}/{log F₁(X)F₂(Y)} and W = C{F₁(X),F₂(Y)}. Using this distribution, an algorithm to generate random variables having bivariate extreme-value distribution is presented. Furthermore, it is shown that for any bivariate extreme-value dependence function C, the distribution of the random variable W = C{ F₁(X),F₂(Y)} belongs to a monoparametric family of distributions. This property is used to derive goodness-of-fit statistics to determine whether a copula belongs to an extreme-value family.