Maximum empirical likelihood estimation of the spectral measure of an extreme-value distribution

Abstract

Consider a random sample from a bivariate distribution function $F$ in the max-domain of attraction of an extreme-value distribution function $G$. This $G$ is characterized by two extreme-value indices and a spectral measure, the latter determining the tail dependence structure of $F$. A major issue in multivariate extreme-value theory is the estimation of the spectral measure ${\Phi }_p$ with respect to the $L_p$ norm. For every $p\in [1,\infty ]$, a nonparametric maximum empirical likelihood estimator is proposed for ${\Phi }_p$. The main novelty is that these estimators are guaranteed to satisfy the moment constraints by which spectral measures are characterized. Asymptotic normality of the estimators is proved under conditions that allow for tail independence. Moreover, the conditions are easily verifiable as we demonstrate through a number of theoretical examples. A simulation study shows a substantially improved performance of the new estimators. Two case studies illustrate how to implement the methods in practice.

Published in at http://dx.doi.org/10.1214/08-AOS677 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Weighted approximations of tail copula processes with application to testing the bivariate extreme value condition

Cited by in corpus (6)

• Statistical Modeling of Spatial Extremes
• A modeler's guide to extreme value software
• Rank-based Estimation under Asymptotic Dependence and Independence, with Applications to Spatial Extremes
• Improved inference on risk measures for univariate extremes
• Rejoinder to "Statistical Modeling of Spatial Extremes"
• Modelling multivariate extreme value distributions via Markov trees