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December 2003 Kernel-type estimators for the extreme value index
P. Groeneboom, H.P. Lopuhaä, P.P. de Wolf
Ann. Statist. 31(6): 1956-1995 (December 2003). DOI: 10.1214/aos/1074290333


A large part of the theory of extreme value index estimation is developed for positive extreme value indices. The best-known estimator of a positive extreme value index is probably the Hill estimator. This estimator belongs to the category of moment estimators, but can also be interpreted as a quasi-maximum likelihood estimator. It has been generalized to a kernel-type estimator, but this kernel-type estimator can, similarly to the Hill estimator, only be used for the estimation of positive extreme value indices. In the present paper, we introduce kernel-type estimators which can be used for estimating the extreme value index over the whole (positive and negative) range. We present a number of results on their distributional behavior and compare their performance with the performance of other estimators, such as moment-type estimators for the whole range and the quasi-maximum likelihood estimator, based on the generalized Pareto distribution. We also discuss an automatic bandwidth selection method and introduce a kernel estimator for a second-order parameter, controlling the speed of convergence.


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P. Groeneboom. H.P. Lopuhaä. P.P. de Wolf. "Kernel-type estimators for the extreme value index." Ann. Statist. 31 (6) 1956 - 1995, December 2003.


Published: December 2003
First available in Project Euclid: 16 January 2004

zbMATH: 1047.62046
MathSciNet: MR2036396
Digital Object Identifier: 10.1214/aos/1074290333

Primary: 60G70 , 62E20 , 62G05
Secondary: 62G20 , 62G30

Keywords: adaptive estimation , extreme value index , second order parameter estimation

Rights: Copyright © 2003 Institute of Mathematical Statistics


Vol.31 • No. 6 • December 2003
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