Estimation of the Mean Vector of a Multivariate Elliptically Contoured Distribution

Abstract

Abtsrcat

This paper deals with the estimation of the mean vector θ of a p-variate elliptically contoured distribution, E_p(θ,Σ,f) based on the sample Y₁Y₂,...,Y_N of size N of size N when it is suspected that for a p× r known matrix B, the hypothesis θ = Bη, η∈ R^r may hold. We consider the following estimators, (i) the unrestricted estimator (UE), (ii) the restricted estimator (RE), (iii) the preliminary test estimator (PTE), (iv) the James—Stein estimator (JSE), and (v) the positive-rule Stein estimator (PRSE). The bias and the risk expressions under the squared loss function are obtained for the five estimators and compared. It is noted that the dominance properties of these estimators remain the same as under normal theory. Further, it is shown that the shrinkage factor of the Stein-type estimators is robust with respect to the mean and unknown mixing distributions.