On empirical Bayes estimation of multivariate regression coefficient

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 anm-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,σIm), 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.