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International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 51695, 18 pages
http://dx.doi.org/10.1155/IJMMS/2006/51695

On empirical Bayes estimation of multivariate regression coefficient

1Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, Canada T6G 2G1
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 Hindawi Publishing Corporation. 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 εiN(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.