Minimum distance estimation of Pickands dependence function for multivariate distributions

Abstract

We consider the problem of estimating the Pickands dependence function corresponding to a multivariate distribution. A minimum distance estimator is proposed which is based on a L2-distance between the logarithms of the empirical and an extreme-value copula. The minimizer can be expressed explicitly as a linear functional of the logarithm of the empirical copula and weak convergence of the corresponding process on the simplex is proved. In contrast to other procedures which have recently been proposed in the literature for the nonparametric estimation of a multivariate Pickands dependence function [see Zhang et al. (2008) and Gudendorf and Segers (2011)], the estimators constructed in this paper do not require knowledge of the marginal distributions and are an alternative to the method which has recently been suggested by Gudendorf and Segers (2012). Moreover, the minimum distance approach allows the construction of a simple test for the hypothesis of a multivariate extreme-value copula, which is consistent against a broad class of alternatives. The finite-sample properties of the estimator and a multiplier bootstrap version of the test are investigated by means of a simulation study.