Empirical Bayes estimation by wavelet series

Abstract

In traditional nonparametric EB (empirical Bayes) setting, the paper proposes generalization of the linear EB estimation method which takes advantage of the flexibility of the wavelet techniques. A nonparametric EB estimator is represented as a wavelet series expansion and the coefficients are estimated by minimizing the prior risk of the estimator. Although wavelet series have been used previously for EB estimation, the method suggested in the paper is completely novel since the EB estimator as a whole is represented as a wavelet series rather than its components. Moreover, the method exploits de-correlating property of wavelets which is not instrumental for the former wavelet-based EB techniques. As a result, estimation of wavelet coefficients requires solution of a well-posed sparse system of linear equations. The technique provides asymptotically optimal EB estimators posterior risks of which tend to zero at the optimal rate as the number of observations tends to infinity.