Pointwise optimality of Bayesian wavelet estimators

Authors: Abramovich, Felix; Angelini, Claudia; Canditiis, Daniela

Source: Annals of the Institute of Statistical Mathematics, Volume 59, Number 3, September 2007 , pp. 425-434(10)

Publisher: Springer

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Abstract:

We consider pointwise mean squared errors of several known Bayesian wavelet estimators, namely, posterior mean, posterior median and Bayes Factor, where the prior imposed on wavelet coefficients is a mixture of an atom of probability zero and a Gaussian density. We show that for the properly chosen hyperparameters of the prior, all the three estimators are (up to a log-factor) asymptotically minimax within any prescribed Besov ball <EquationSource Format="TEX"><![CDATA[$$B^{s}_{p},_{q} (M)$$]]></EquationSource> . We discuss the Bayesian paradox and compare the results for the pointwise squared risk with those for the global mean squared error.

Keywords: Bayes Factor; Bayes model; Bayesian paradox; Besov spaces; Minimax rates; Nonparametric regression; Point estimation; Posterior mean; Posterior median; Wavelets

Document Type: Research article

DOI: http://dx.doi.org/10.1007/s10463-006-0071-7

Affiliations: 1: Email: felix@post.tau.ac.il

Publication date: 2007-09-01