Wavelet deconvolution

Author

Fan, Jianqing ; Koo, Ja-Yong

Author_Institution

Dept. of Stat., Chinese Univ. of Hong Kong, Shatin, China

Volume

48

Issue

3

fYear

2002

fDate

3/1/2002 12:00:00 AM

Firstpage

734

Lastpage

747

Abstract

This paper studies the issue of optimal deconvolution density estimation using wavelets. The approach taken here can be considered as orthogonal series estimation in the more general context of the density estimation. We explore the asymptotic properties of estimators based on thresholding of estimated wavelet coefficients. Minimax rates of convergence under the integrated square loss are studied over Besov classes B_σpq of functions for both ordinary smooth and supersmooth convolution kernels. The minimax rates of convergence depend on the smoothness of functions to be deconvolved and the decay rate of the characteristic function of convolution kernels. It is shown that no linear deconvolution estimators can achieve the optimal rates of convergence in the Besov spaces with p<2 when the convolution kernel is ordinary smooth and super smooth. If the convolution kernel is ordinary smooth, then linear estimators can be improved by using thresholding wavelet deconvolution estimators which are asymptotically minimax within logarithmic terms. Adaptive minimax properties of thresholding wavelet deconvolution estimators are also discussed

Keywords

adaptive estimation; convergence of numerical methods; deconvolution; minimax techniques; nonlinear estimation; parameter estimation; wavelet transforms; Besov functions; Besov spaces; adaptive minimax properties; asymptotic properties; characteristic function; convolution kernels; decay rate; estimated wavelet coefficients thresholding; integrated square loss; linear deconvolution estimators; minimax convergence rates; nonlinear wavelet deconvolution estimator; optimal deconvolution density estimation; ordinary smooth convolution kernels; orthogonal series estimation; supersmooth convolution kernels; wavelet deconvolution estimators; Convergence; Convolution; Deconvolution; Density functional theory; Inverse problems; Kernel; Minimax techniques; Random variables; Statistics; Wavelet coefficients;

fLanguage

English

Journal_Title

Information Theory, IEEE Transactions on

Publisher

ieee

ISSN

0018-9448

Type

jour

DOI

10.1109/18.986021

Filename

986021