Wavelet Optimal Estimations for Density Functions under Severely Ill-Posed Noises
Motivated by Lounici and Nickl's work (2011), this paper considers the problem of estimation of a density f based on an independent and identically distributed sample Y1,…,Yn from g=f*φ. We show a wavelet optimal estimation for a density (function) over Besov ball Br,qs(L) and Lp risk (1≤p<...
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| Format: | Article |
| Language: | English |
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Wiley
2013-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2013/260573 |
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| author | Rui Li Youming Liu |
| author_facet | Rui Li Youming Liu |
| author_sort | Rui Li |
| collection | DOAJ |
| description | Motivated by Lounici and Nickl's work (2011), this paper considers the problem of estimation of a density f based on an independent and identically distributed sample Y1,…,Yn
from g=f*φ. We show a wavelet optimal estimation for a density (function) over Besov ball Br,qs(L) and Lp risk (1≤p<∞) 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). |
| format | Article |
| id | doaj-art-b7889df20ce646e08a4d75d8fbffeeff |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-b7889df20ce646e08a4d75d8fbffeeff2025-02-03T06:42:15ZengWileyAbstract and Applied Analysis1085-33751687-04092013-01-01201310.1155/2013/260573260573Wavelet Optimal Estimations for Density Functions under Severely Ill-Posed NoisesRui Li0Youming Liu1Department of Applied Mathematics, Beijing University of Technology, Pingle Yuan 100, Beijing 100124, ChinaDepartment of Applied Mathematics, Beijing University of Technology, Pingle Yuan 100, Beijing 100124, ChinaMotivated by Lounici and Nickl's work (2011), this paper considers the problem of estimation of a density f based on an independent and identically distributed sample Y1,…,Yn from g=f*φ. We show a wavelet optimal estimation for a density (function) over Besov ball Br,qs(L) and Lp risk (1≤p<∞) 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).http://dx.doi.org/10.1155/2013/260573 |
| spellingShingle | Rui Li Youming Liu Wavelet Optimal Estimations for Density Functions under Severely Ill-Posed Noises Abstract and Applied Analysis |
| title | Wavelet Optimal Estimations for Density Functions under Severely Ill-Posed Noises |
| title_full | Wavelet Optimal Estimations for Density Functions under Severely Ill-Posed Noises |
| title_fullStr | Wavelet Optimal Estimations for Density Functions under Severely Ill-Posed Noises |
| title_full_unstemmed | Wavelet Optimal Estimations for Density Functions under Severely Ill-Posed Noises |
| title_short | Wavelet Optimal Estimations for Density Functions under Severely Ill-Posed Noises |
| title_sort | wavelet optimal estimations for density functions under severely ill posed noises |
| url | http://dx.doi.org/10.1155/2013/260573 |
| work_keys_str_mv | AT ruili waveletoptimalestimationsfordensityfunctionsunderseverelyillposednoises AT youmingliu waveletoptimalestimationsfordensityfunctionsunderseverelyillposednoises |