Statistical properties of couples of bivariate extreme-value copulas

Let (X,Y) be a bivariate random vector whose distribution function H(x,y) belongs to the class of bivariate extreme-value distributions. If F(1) and F(2) are the marginals of X and Y, then H(x,y) = C{F(1)(x),F(2)(y)}, where C is a bivariate extreme-value dependence function. This paper gives the joint distribution of the random variables Z = {log F(1)(X)}/{log F(1)(X)F(2)(Y)} and W = C{F(1)(X),F(2)(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{F(1)(X),F(2)(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.