International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Issue 13, Article ID 51695, 18 pages
doi:10.1155/IJMMS/2006/51695

On empirical Bayes estimation of multivariate regression coefficient

R.J. Karunamuni1 and L. Wei2

1Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton T6G 2G1, AB, Canada
 2Department of Statistics and Finance, University of Science and Technology of China, Hefei, Anhui 230026, China

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.