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Biometrika 2006 93(3):705-722; doi:10.1093/biomet/93.3.705
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© 2006 Biometrika Trust

Empirical Bayes block shrinkage of wavelet coefficients via the noncentral {chi}2 distribution

Xue Wang1 and Andrew T. A. Wood2

1 Department of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, U.K. xue.wang{at}bristol.ac.uk, 2 School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, U.K. atw{at}maths.nott.ac.uk

Empirical Bayes approaches to the shrinkage of empirical wavelet coefficients have generated considerable interest in recent years. Much of the work to date has focussed on shrinkage of individual wavelet coefficients in isolation. In this paper we propose an empirical Bayes approach to simultaneous shrinkage of wavelet coefficients in a block, based on the block sum of squares. Our approach exploits a useful identity satisfied by the noncentral {chi}2 density and provides some tractable Bayesian block shrinkage procedures. Our numerical results indicate that the new procedures perform very well.

Key Words: Block size; Heavy-tailed prior; Nonparametric regression; Posterior mean; Posterior median; Wavelet thresholding.

Received January 2003. Revised February 2006.


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