General maximum likelihood empirical Bayes estimation of normal means

TL;DR

This paper introduces a general maximum likelihood empirical Bayes method for estimating normal means, demonstrating near-optimality and superior performance over existing estimators through theoretical proofs and simulations.

Contribution

The paper develops a novel GMLEB approach that is theoretically near-minimax and empirically outperforms traditional estimators like James--Stein in various settings.

Findings

GMLEB achieves near-minimax mean squared error.

GMLEB outperforms James--Stein and threshold estimators in simulations.

Method is effective under mild moment conditions on the means.

Abstract

We propose a general maximum likelihood empirical Bayes (GMLEB) method for the estimation of a mean vector based on observations with i.i.d. normal errors. We prove that under mild moment conditions on the unknown means, the average mean squared error (MSE) of the GMLEB is within an infinitesimal fraction of the minimum average MSE among all separable estimators which use a single deterministic estimating function on individual observations, provided that the risk is of greater order than $(\log n)^5/n$. We also prove that the GMLEB is uniformly approximately minimax in regular and weak $\ell_p$ balls when the order of the length-normalized norm of the unknown means is between $(\log n)^{\kappa_1}/n^{1/(p\wedge2)}$ and $n/(\log n)^{\kappa_2}$. Simulation experiments demonstrate that the GMLEB outperforms the James--Stein and several state-of-the-art threshold estimators in a wide range of settings without much down side.