The Annals of Statistics

Minimax Risk Bounds in Extreme Value Theory

Holger Drees

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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.

Article information

Source
Ann. Statist., Volume 29, Number 1 (2001), 266-294.

Dates
First available in Project Euclid: 5 August 2001

Permanent link to this document
https://projecteuclid.org/euclid.aos/996986509

Digital Object Identifier
doi:10.1214/aos/996986509

Mathematical Reviews number (MathSciNet)
MR1833966

Zentralblatt MATH identifier
1029.62046

Subjects
Primary: 62C20: Minimax procedures 62G32: Statistics of extreme values; tail inference
Secondary: 62G05: Estimation 62G15: Tolerance and confidence regions

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

Citation

Drees, Holger. Minimax Risk Bounds in Extreme Value Theory. Ann. Statist. 29 (2001), no. 1, 266--294. doi:10.1214/aos/996986509. https://projecteuclid.org/euclid.aos/996986509


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