Modulation of estimators and confidence sets

• 成果类型: Article
• 署名作者: Beran, R; Dümbgen, L
• 署名单位: University of California System; University of California Berkeley; Ruprecht Karls University Heidelberg
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• 发表日期: 1998
• 页码: 1826-1856
• 关键词: Nonparametric regression; Metric Entropy; cp

摘要

An unknown signal plus white noise is observed at n discrete time points. Within a large convex class of linear estimators of xi, we choose the estimator <(xi)over cap> that minimizes estimated quadratic risk. By construction, <(xi)over cap> is nonlinear. This estimation is done after orthogonal transformation of the data to a reasonable coordinate system. The procedure adaptively tapers the coefficients of the transformed data. If the class of candidate estimators satisfies a uniform entropy condition, then <(xi)over cap> is asymptotically minimax in Pinsker's sense over certain ellipsoids in the parameter space and shares one such asymptotic minimax property with the James-Stein estimator. We describe computational algorithms for <(xi)over cap> and construct confidence sets for the unknown signal. These confidence sets are centered at <(xi)over cap>, have correct asymptotic coverage probability and have relatively small risk as set-valued estimators of xi.