Adaptive estimation of and oracle inequalities for probability densities and characteristic functions

• 成果类型: Article
• 署名作者: Efromovich, Sam
• 署名单位: University of Texas System; University of Texas Dallas
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/009053607000000965
• 发表日期: 2008
• 页码: 1127-1155
• 关键词: nonparametric-estimation; wavelet; EQUIVALENCE; smoothness; regression; shrinkage

摘要

The theory of adaptive estimation and oracle inequalities for the case of Gaussian-shift-finite-interval experiments has made significant progress in recent years. In particular, sharp-minimax adaptive estimators and exact exponential-type oracle inequalities have been suggested for a vast set of functions including analytic and Sobolev with any positive index as well as for Efromovich-Pinsker and Stein blockwise-shrinkage estimators. Is it possible to obtain similar results for a more interesting applied problem of density estimation and/or the dual problem of characteristic function estimation? The answer is yes. In particular, the obtained results include exact exponential-type oracle inequalities which allow to consider, for the first time in the literature, a simultaneous sharp-minimax estimation of Sobolev densities with any positive index (not necessarily larger than 1/2), infinitely differentiable densities (including analytic, entire and stable), as well as of not absolutely integrable characteristic functions. The same adaptive estimator is also rate minimax over a familiar class of distributions with bounded spectrum where the density and the characteristic function can be estimated with the parametric rate.