On empirical Bayes estimation of multivariate regression coefficient

Abstract

We investigate the empirical Bayes estimation problem of multivariate regression coefficients under squared error loss function. In particular, we consider the regression model Y=Xβ+ε, where Y is an m-vector of observations, X is a known m×k matrix, β is an unknown k-vector, and ε is an m-vector of unobservable random variables. The problem is squared error loss estimation of β based on some “previous” data Y1,…,Yn as well as the “current” data vector Y when β is distributed according to some unknown distribution G, where Yi satisfies Yi=Xβi+εi, i=1,…,n. We construct a new empirical Bayes estimator of β when εi∼N(0,σ2Im), i=1,…,n. The performance of the proposed empirical Bayes estimator is measured using the mean squared error. The rates of convergence of the mean squared error are obtained.