The Annals of Statistics

Minimax Risk Bounds in Extreme Value Theory

Holger Drees

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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

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

First available in Project Euclid: 5 August 2001

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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

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