NEW ESTIMATORS OF THE PICKANDS DEPENDENCE FUNCTION AND A TEST FOR EXTREME-VALUE DEPENDENCE

• 成果类型: Article
• 署名作者: Buecher, Axel; Dette, Holger; Volgushev, Stanislav
• 署名单位: Ruhr University Bochum
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/11-AOS890
• 发表日期: 2011
• 页码: 1963-2006
• 关键词: value distributions; nonparametric-estimation; value copulas; models

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.