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

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

2011-02-02

arxiv.org/abs/1102.0405v2

Annals of Statistics 2011, Vol. 39, No. 4, 1963-2006

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.1214/11-AOS890 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of

Scientific paper

10.1214/11-AOS890

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.

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