Empirical Bayes approach to block wavelet function estimation

Abstract

Wavelet methods have demonstrated considerable success in function estimation through term-by-term thresholding of the empirical wavelet coefficients. However, it has been shown that grouping the empirical wavelet coefficients into blocks and making simultaneous threshold decisions about all the coefficients in each block has a number of advantages over term-by-term wavelet thresholding, including asymptotic optimality and better mean squared error performance in finite sample situations. An empirical Bayes approach to incorporating information on neighbouring empirical wavelet coefficients into function estimation that results in block wavelet shrinkage and block wavelet thresholding estimators is considered. Simulated examples are used to illustrate the performance of the resulting estimators, and to compare these estimators with several existing non-Bayesian block wavelet thresholding estimators. It is observed that the proposed empirical Bayes block wavelet shrinkage and block wavelet thresholding estimators outperform the non-Bayesian block wavelet thresholding estimators in finite sample situations. An application to a data set that was collected in an anaesthesiological study is also presented.