Minimax risk bounds tn extreme value theory

• 成果类型: Article
• 署名作者: Drees, H
• 署名单位: Ruprecht Karls University Heidelberg
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/aos/996986509
• 发表日期: 2001
• 页码: 266-294
• 关键词: order-statistics; value index; confidence-intervals; geometrizing rates; CONVERGENCE; tail; inference

摘要

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.