Testing for Tail Independence in Extreme Value models

Let (X,Y) be a random vector which follows in its upper tail a bivariate extreme value distribution with reverse exponentialmargins. We show that the conditional distribution function (df) of X + Y, given that X + Y>c, converges to the df F (t) = t ², , as if and only if X,Y are tail independent. Otherwise, the limit is F (t) = t. This is utilized to test for the tail independence of X, Y via various tests, including the one suggested by the Neyman–Pearson lemma. Simulations show that the Neyman–Pearson test performs best if the threshold c is close to 0, whereas otherwise it is the Kolmogorov–Smirnov test that performs best. The mathematical conditions are studied under which the Neyman–Pearson approach actually controls the type I error. Our considerations are extended to extreme value distributions in arbitrary dimensions as well as to distributions which are in a differentiable spectral neighborhood of an extreme value distribution.