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February 2001 Minimax Risk Bounds in Extreme Value Theory
Holger Drees
Ann. Statist. 29(1): 266-294 (February 2001). DOI: 10.1214/aos/996986509

Abstract

Asymptotic minimax risk bounds for estimators of a positive extreme value index under zero-one loss are investigated in the classical i.i.d. setup. To this end, we prove the weak convergence of suitable local experiments with Pareto distributions as center of localization to a white noise model, which was previously studied in the context of nonparametric local density estimation and regression. From this result we derive upper and lower bounds on the asymptotic minimax risk in the local and in certain global models as well. Finally, the implications for fixed-length confidence intervals are discussed. In particular, asymptotic confidence intervals with almost minimal length are constructed, while the popular Hill estimator is shown to yield a little longer confidence intervals.

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Holger Drees. "Minimax Risk Bounds in Extreme Value Theory." Ann. Statist. 29 (1) 266 - 294, February 2001. https://doi.org/10.1214/aos/996986509

Information

Published: February 2001
First available in Project Euclid: 5 August 2001

zbMATH: 1029.62046
MathSciNet: MR1833966
Digital Object Identifier: 10.1214/aos/996986509

Subjects:
Primary: 62C20 , 62G32
Secondary: 62G05 , 62G15

Keywords: confidence intervals , convergence of experiments , extreme value index , Gaussian shift , Hill estimator , local experiment , minimax affine estimator , White noise , zero-one loss

Rights: Copyright © 2001 Institute of Mathematical Statistics

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Vol.29 • No. 1 • February 2001
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