Nonparametric estimation of the dependence function for a multivariate extreme value distribution

Abstract

Understanding and modeling dependence structures for multivariate extreme values are of interest in a number of application areas. One of the well-known approaches is to investigate the Pickands dependence function. In the bivariate setting, there exist several estimators for estimating the Pickands dependence function which assume known marginal distributions [J. Pickands, Multivariate extreme value distributions, Bull. Internat. Statist. Inst., 49 (1981) 859–878; P. Deheuvels, On the limiting behavior of the Pickands estimator for bivariate extreme-value distributions, Statist. Probab. Lett. 12 (1991) 429–439; P. Hall, N. Tajvidi, Distribution and dependence-function estimation for bivariate extreme-value distributions, Bernoulli 6 (2000) 835–844; P. Capéraà, A.-L. Fougères, C. Genest, A nonparametric estimation procedure for bivariate extreme value copulas, Biometrika 84 (1997) 567–577]. In this paper, we generalize the bivariate results to p-variate multivariate extreme value distributions with p ⩾ 2 . We demonstrate that the proposed estimators are consistent and asymptotically normal as well as have excellent small sample behavior.