New estimators of the Pickands dependence function and a test for extreme-value dependence
Axel Bücher, Holger Dette, Stanislav Volgushev
The Annals of Statistics · 2011 · Volume 39 · Issue 4
Abstract
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 L2-distance between the logarithm of the copula C(y1−t, yt) 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.
Keywords
60G32, 62G05, 62G20, empirical copula process, Extreme-value copula, minimum distance estimation, Pickands dependence function, test for extreme-value dependence, weak convergence
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