On Optimality of Bayesian Wavelet Estimators

• DOI: 10.1111/j.1467-9469.2004.02-087.x
• Publication Date: 2004-05-19T11:18:05Z
• Authors: Felix Abramovich; Umberto Amato; Claudia Angelini
• Keywords: Bayes Factor; Besov spaces; 0502 economics and business; 05 social sciences; Minimax estimation; Nonparametric regression; Wavelets; 0101 mathematics; 01 natural sciences

Abstract. 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 (2s+2)/(2s+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.