Nonparametric empirical Bayes estimation based on generalized Laguerre series

TL;DR

This paper introduces a nonparametric empirical Bayes method using generalized Laguerre series to estimate the Bayes estimator for distributions supported on positive real numbers, achieving asymptotic optimality.

Contribution

It develops a novel Laguerre series-based estimator for empirical Bayes problems with positive support, including a method for selecting parameters to ensure finite variance.

Findings

Estimator is asymptotically minimax optimal.

Provides a strategy for parameter selection for finite variance.

Shows convergence rate comparison with existing methods.

Abstract

In this work, we delve into the nonparametric empirical Bayes theory and approximate the classical Bayes estimator by a truncation of the generalized Laguerre series and then estimate its coefficients by minimizing the prior risk of the estimator. The minimization process yields a system of linear equations the size of which is equal to the truncation level. We focus on the empirical Bayes estimation problem when the mixing distribution, and therefore the prior distribution, has a support on the positive real half-line or a subinterval of it. By investigating several common mixing distributions, we develop a strategy on how to select the parameter of the generalized Laguerre function basis so that our estimator {possesses a finite} variance. We show that our generalized Laguerre empirical Bayes approach is asymptotically optimal in the minimax sense. Finally, our convergence rate is compared and contrasted with {several} results from the literature.