Wavelet Shrinkage for Nonparametric Mixed-Effects Models

In this article, a nonparametric regression problem is discussed on wavelet bases via a Bayesian structure. The resulting model is a nonparametric mixed-effects model. Regularities of the prior and the posterior spaces are studied and compared. Both Bayes and empirical Bayes estimators are derived. Moreover, the proposed empirical Bayes estimator is shown to have the Gauss-Markov type optimality when the prior parameters are available. It is also equivalent to a Sobolev regularization. Adaptive variants of the empirical Bayes estimator are also investigated when the prior parameters are not feasible. The theoretical justifications and simulation results of these new wavelet shrinkage methods are reported.