Supersmooth density estimations over L^p risk by wavelets

作     者：LI Rui LIU YouMing 

作者机构：Department of Applied Mathematics Beijing University of Technology College of Science Tianjin University of Technology 

出 版 物：《Science China Mathematics》 (中国科学：数学（英文版）)

年 卷 期：2017年第60卷第10期

页      面：1901-1922页

核心收录：

学科分类：02[经济学] 0202[经济学-应用经济学] 020208[经济学-统计学] 07[理学] 0714[理学-统计学（可授理学、经济学学位）] 070103[理学-概率论与数理统计] 0701[理学-数学] 

基　　金：supported by National Natural Science Foundation of China (Grant Nos. 11526150  11601383 and 11271038) 

主　　题：wavelet estimation supersmooth density additive noise optimality 

摘      要：This paper studies wavelet estimations for supersmooth density functions with additive noises. We first show lower bounds of Lprisk(1 p ∞) with both moderately and severely ill-posed noises. Then a Shannon wavelet estimator provides optimal or nearly-optimal estimations over Lprisks for p 2, and a nearly-optimal result for 1 p 2 under both noises. In the nearly-optimal cases, the ratios of upper and lower bounds are determined. When p = 1, we give a nearly-optimal estimation with moderately ill-posed noise by using the Meyer wavelet. Finally, the practical estimators are considered. Our results are motivated by the work of Pensky and Vidakovic(1999), Butucea and Tsybakov(2008), Comte et al.(2006), Lacour(2006) and Lounici and Nickl(2011).