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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 $$B^{s}_{p},_{q} (M)$$ . We discuss the Bayesian paradox and compare the results for the pointwise squared risk with those for the global mean squared error. [ABSTRACT FROM AUTHOR]

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 χ2 density and provides some tractable Bayesian block shrinkage procedures. Our numerical results indicate that the new procedures perform very well. [ABSTRACT FROM AUTHOR]

We investigate the asymptotic optimality of several Bayesian wavelet estimators, namely, posterior mean, posterior median and Bayes Factor, where the prior imposed on wavelet coefficients is a mixture of a mass function at zero and a Gaussian density. We show that in terms of the mean squared error, for the properly chosen hyperparameters of the prior, all the three resulting Bayesian wavelet estimators achieve optimal minimax rates within any prescribed Besov space for p ≥ 2. For 1 ≤ p < 2, the Bayes Factor is still optimal for (2 s+2)/(2 s+1) ≤ p < 2 and always outperforms the posterior mean and the posterior median that can achieve only the best possible rates for linear estimators in this case. [ABSTRACT FROM AUTHOR]