New estimators of the pickands dependence function and a test for extreme-value dependence

摘要

We propose a new class of estimators for Pickands dependence function which is based on the concept of minimum distance estimation. An explicit integral representation of the function A * (t), which minimizes a weighted L ~2-distance between the logarithm of the copula C(y ~(1?t), y ~t) and functions of the form A(t) log(y) is derived. If the unknown copula is an extreme-value copula, the function A ~* (t) coincides with Pickands dependence function. Moreover, even if this is not the case, the function A ~* (t) always satisfies the boundary conditions of a Pickands dependence function. The estimators are obtained by replacing the unknown copula by its empirical counterpart and weak convergence of the corresponding process is shown. A comparison with the commonly used estimators is performed from a theoretical point of view and by means of a simulation study. Our asymptotic and numerical results indicate that some of the new estimators outperform the estimators, which were recently proposed by Genest and Segers [Ann. Statist. 37 (2009) 2990-3022]. As a by-product of our results, we obtain a simple test for the hypothesis of an extreme-value copula, which is consistent against all positive quadrant dependent alternatives satisfying weak differentiability assumptions of first order.