Estimation of the mean vector of a multivariate normal distribution: subspace hypothesis

Abstract

This paper considers the estimation of the mean vector θ of a p-variate normal distribution with unknown covariance matrix Σ when it is suspected that for a p × r known matrix B the hypothesis θ = B η , η ∈ R r may hold. We consider empirical Bayes estimators which includes (i) the unrestricted unbiased (UE) estimator, namely, the sample mean vector (ii) the restricted estimator (RE) which is obtained when the hypothesis θ = B η holds (iii) the preliminary test estimator (PTE), (iv) the James–Stein estimator (JSE), and (v) the positive-rule Stein estimator (PRSE). The biases and the risks under the squared loss function are evaluated for all the five estimators and compared. The numerical computations show that PRSE is the best among all the five estimators even when the hypothesis θ = B η is true.