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 = X β + 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 Y 1 , … , Y n as well as the “current†data vector Y when β is distributed according to some unknown distribution G , where Y i satisfies Y i = X β i + ε i , i = 1 , … , n . We construct a new empirical Bayes estimator of β when ε i ∼ N ( 0 , σ 2 I m ) , 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.