Open Access
October 2009 Maximum empirical likelihood estimation of the spectral measure of an extreme-value distribution
John H. J. Einmahl, Johan Segers
Ann. Statist. 37(5B): 2953-2989 (October 2009). DOI: 10.1214/08-AOS677


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 Φp with respect to the Lp norm. For every p∈[1, ∞], a nonparametric maximum empirical likelihood estimator is proposed for Φp. 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.


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John H. J. Einmahl. Johan Segers. "Maximum empirical likelihood estimation of the spectral measure of an extreme-value distribution." Ann. Statist. 37 (5B) 2953 - 2989, October 2009.


Published: October 2009
First available in Project Euclid: 17 July 2009

zbMATH: 1173.62042
MathSciNet: MR2541452
Digital Object Identifier: 10.1214/08-AOS677

Primary: 62G05 , 62G30 , 62G32
Secondary: 60F05 , 60F17 , 60G70

Keywords: functional central limit theorem , local empirical process , moment constraint , multivariate extremes , National Health and Nutrition Examination Survey , nonparametric maximum likelihood estimator , tail dependence

Rights: Copyright © 2009 Institute of Mathematical Statistics

Vol.37 • No. 5B • October 2009
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