Supersmooth density estimations over L^p risk by wavelets  

作　　者：LI Rui LIU YouMing 

机构地区：[1]Department of Applied Mathematics, Beijing University of Technology [2]College of Science, Tianjin University of Technology

出　　处：《Science China Mathematics》2017年第10期1901-1922,共22页中国科学：数学（英文版）

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

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).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 L^p risks 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）.

关 键 词：wavelet estimation supersmooth density additive noise OPTIMALITY 

分 类 号：O212[理学—概率论与数理统计]