Minimum distance estimators of the Pickands dependence function and related tests of multivariate extreme-value dependence

Résumé

We consider the problem of estimating the Pickands dependence function corresponding to a multivariate extreme-value distribution. A minimum distance estimator is proposed which is based on an $L^2$-distance between the logarithms of the empirical and the unknown 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 in [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.