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Minimax risk bounds in extreme value theory. (English) Zbl 1029.62046

Summary: 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.

MSC:

62G32 Statistics of extreme values; tail inference
62G05 Nonparametric estimation
62G15 Nonparametric tolerance and confidence regions
62C20 Minimax procedures in statistical decision theory
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