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February 1998 Optimal rates of convergence for estimates of the extreme value index
Holger Drees
Ann. Statist. 26(1): 434-448 (February 1998). DOI: 10.1214/aos/1030563992

Abstract

Hall and Welsh established the best attainable rate of convergence for estimates of a positive extreme value index $\gamma$ under a certain second order condition implying that the distribution function of the maximum of n random variables converges at an algebraic rate to the pertaining extreme value distribution. As a first generalization, we obtain a surprisingly sharp bound on the estimation error if $\gamma$ is still assumed to be positive, but the rate of convergence of the maximum may be nonalgebraic. This result allows a more accurate evaluation of the asymptotic performance of an estimator for $\gamma$ than the Hall and Welsh theorem. For example, it is proved that the Hill and the Pickands estimators achieve the optimal rate, but only the Hill estimator attains the sharp bound. Finally, an analogous result is derived for a general, not necessarily positive, extreme value index. In this situation it turns out that location invariant estimators show the best performance.

Citation

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Holger Drees. "Optimal rates of convergence for estimates of the extreme value index." Ann. Statist. 26 (1) 434 - 448, February 1998. https://doi.org/10.1214/aos/1030563992

Information

Published: February 1998
First available in Project Euclid: 28 August 2002

zbMATH: 0934.62047
MathSciNet: MR1608148
Digital Object Identifier: 10.1214/aos/1030563992

Subjects:
Primary: 62G05
Secondary: 62G20 , 62G35

Keywords: extreme value index , Hill estimator , moment estimator , optimality , Pickands estimator , rate of convergence , robustness

Rights: Copyright © 1998 Institute of Mathematical Statistics

Vol.26 • No. 1 • February 1998
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