International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 51695, 18 pages
doi:10.1155/IJMMS/2006/51695
On empirical Bayes estimation of multivariate regression
coefficient
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
Received 11 November 2005; Revised 19 April 2006; Accepted 4 May 2006
Copyright © 2006 R. J. Karunamuni and L. Wei. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.