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

TL;DR

This paper introduces new estimators for the Pickands dependence function based on minimum distance estimation, providing a test for extreme-value dependence with improved performance demonstrated through theoretical analysis and simulations.

Contribution

The paper develops a novel class of estimators for the Pickands dependence function using integral representations and minimum distance, along with a new test for extreme-value dependence.

Findings

New estimators outperform existing ones in simulations.

The proposed test is consistent against all positive quadrant dependent alternatives.

Weak convergence of the estimators is established.

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 $L^2$-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.