Adaptive Nonparametric Empirical Bayes Estimation Via Wavelet Series: the Minimax Study

TL;DR

This paper develops an adaptive wavelet-based nonparametric empirical Bayes estimator that achieves minimax optimality by estimating wavelet coefficients through a sparse linear system and adaptively selecting resolution levels.

Contribution

It introduces a novel wavelet series approach for empirical Bayes estimation with adaptive resolution selection, achieving minimax optimal convergence rates.

Findings

Estimator attains asymptotic optimality.

Method is computationally efficient.

Provides numerous practical examples.

Abstract

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 (1997) method. The method is computationally efficient and provides asymptotically optimal adaptive EB estimators. The theory is supplemented by numerous examples.