Adaptive nonparametric empirical Bayes estimation via wavelet series: The minimax study

• Lower bounds for the risk of the nonparametric empirical Bayes estimators are derived for a general conditional distribution.
• Wavelet series empirical Bayes estimator is constructed.
• Adaptive choice of resolution level by Lepski et al. (1997) method.
• Numerous examples of construction of empirical Bayes estimators, and the lower and the upper bounds for the risk.

In the present paper, we derive lower bounds for the risk of the nonparametric empirical Bayes estimators. In order to attain the optimal convergence rate, we propose generalization of the linear empirical Bayes estimation method which takes advantage of the flexibility of the wavelet techniques. We present an empirical Bayes estimator as a wavelet series expansion and estimate coefficients by minimizing the prior risk of the estimator. As a result, estimation of wavelet coefficients requires solution of a well-posed low-dimensional sparse system of linear equations. The dimension of the system depends on the size of wavelet support and smoothness of the Bayes estimator. An adaptive choice of the resolution level is carried out using Lepski et al. (1997) method. The method is computationally efficient and provides asymptotically optimal adaptive EB estimators. The theory is supplemented by numerous examples.