On the distribution of Pickands coordinates in bivariate EV and GP models

Let (U, V) be a random vector with U <= 0, V <= 0. The random variables Z = V/(U+ V), C = U + V are the Pickands coordinates of (U, V). They are a useful tool for the investigation of the tail behaviour in bivariate peaks-over-threshold models in extreme value theory. We compute the distribution of (Z, C) among others under the assumption that the distribution function H of (U, V) is in a smooth neighborhood of a generalized Pareto distribution (GP) with uniform marginals. It turns out that if H is a GP, then Z and C are independent, conditional on C > c >= - 1. These results are used to derive approximations of the empirical point process of the exceedances (Z(i), C-i) with C-i > c in an iid sample of size n. Local asymptotic normality is established for the approximating point process in a parametric model, where c = c(n) TO as n -> infinity . (c) 2004 Elsevier Inc. All rights reserved.