Consistency of Bayesian Estimates for the Sum of Squared Normal Means with a Normal Prior

Abstract

We consider the problem of estimating the sum of squared means when the data (x ₁,...,x _n ) are independent values with x _i  ∼ N(θ _i , 1) and θ ₁,  θ ₂... are a priori i.i.d. N(0, σ ²) with σ ² known. This example has posed difficulties for many approaches to inference. We examine the consistency properties of several estimators derived from Bayesian considerations. We prove that a particular Bayesian estimate (LRSE) is consistent in a wider set of circumstances than other Bayesian estimates like the posterior mean and mode. We show that the LRSE is either equal to the positive part of the UMVUE or differs from it with a relative error no greater than 2/n. We also prove a consistency result for interval estimation and discuss checking for prior-data conflict. While it can be argued that the choice of the N(0,σ ²) prior is inappropriate when σ ² is chosen large to reflect noninformativity, this argument is not applicable when σ ² is chosen to reflect knowledge about the unknowns. As such it is important to show that there are consistent Bayesian estimation procedures using this prior.