Wavelet deconvolution

Jianqing Fan, Ja Yong Koo

Research output: Contribution to journalArticle

102 Scopus citations

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 Bσpq 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.

Original languageEnglish (US)
Pages (from-to)734-747
Number of pages14
JournalIEEE Transactions on Information Theory
Volume48
Issue number3
DOIs
StatePublished - Mar 1 2002
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences

Keywords

  • Adaptive estimation
  • Besov spaces
  • Kullback-Leibler information
  • Linear estimators
  • Minimax estimation
  • Thresholding
  • Wavelet bases

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