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Abstract and Applied Analysis
Volume 2013, Article ID 260573, 7 pages
Research Article

Wavelet Optimal Estimations for Density Functions under Severely Ill-Posed Noises

Department of Applied Mathematics, Beijing University of Technology, Pingle Yuan 100, Beijing 100124, China

Received 6 October 2013; Accepted 13 November 2013

Academic Editor: Ding-Xuan Zhou

Copyright © 2013 Rui Li and Youming Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Motivated by Lounici and Nickl's work (2011), this paper considers the problem of estimation of a density based on an independent and identically distributed sample from . We show a wavelet optimal estimation for a density (function) over Besov ball and risk () in the presence of severely ill-posed noises. A wavelet linear estimation is firstly presented. Then, we prove a lower bound, which shows our wavelet estimator optimal. In other words, nonlinear wavelet estimations are not needed in that case. It turns out that our results extend some theorems of Pensky and Vidakovic (1999), as well as Fan and Koo (2002).