Robust Empirical Bayes Estimation of the Elliptically Countoured Covariance Matrix

Abstract

Let S be the matrix of residual sum of square in linear model Y = Aβ + e, where the matrix of errors is distributed as elliptically contoured with unknown scale matrix Σ. For Stein loss function, L1(Σˆ,Σ) = tr(ΣˆΣ−1)−log|ΣˆΣ−1|−p, and squared loss function, L2 (Σˆ , Σ) = tr(Σˆ Σ−1 − I)2 , we offer empirical Bayes estimators of Σ, which dominate any scalar multiple of S, i.e., aS, by an effective amount. In fact, this study somehow shows that improvement of the empirical Bayes estimators obtained under the normality assumption remains robust under elliptically contoured model.