Wavelet optimal estimations for density functions under severely ill-posed noises
type: Item
instance of: scholarly article
title: Wavelet optimal estimations for density functions under severely ill-posed noises
zbMATH Open document ID: 1306.62095
DOI: 10.1155/2013/260573
author: Rui Li
author: Youming Liu
published in: Abstract and Applied Analysis
publication date: 2014-06-23T00:00:00Z
review text: Summary: Motivated by \textit{K. Lounici} and \textit{R. Nickl}'s work [Ann. Stat. 39, No. 1, 201--231 (2011; Zbl 1209.62060)], this paper considers the problem of estimation of a density \(f\) based on an independent and identically distributed sample \(Y_1,\dots,Y_n\) from \(g=f\ast\varphi\). We show a wavelet optimal estimation for a density (function) over Besov ball \(B^s_{r,q}(L)\) and \(L^p\) risk (\(1\leq p<\infty\)) 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 \textit{M. Pensky} and \textit{B. Vidakovic} [Ann. Stat. 27, No. 6, 2033--2053 (1999; Zbl 0962.62030)], as well as \textit{J. Fan} and \textit{J.-Y. Koo} [IEEE Trans. Inf. Theory 48, No. 3, 734--747 (2002; Zbl 1071.94511)].
Mathematics Subject Classification ID: 62G07
Mathematics Subject Classification ID: 62G05
Mathematics Subject Classification ID: 42C40
zbMATH DE Number: 6306569
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2013/260573
OpenAlex ID: W2088099074
Wikidata QID: Q58915705
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description: scientific article; zbMATH DE number 6306569