Estimation of a Normal Covariance Matrix with Incomplete Data under Stein′s Loss

Abstract

Suppose that we have (n − a) independent observations from Np(0, Σ) and that, in addition, we have a independent observations available on the last (p − c) coordinates. Assuming that both observations are independent, we consider the problem of estimating Σ under the Stein′s loss function, and show that some estimators invariant under the permutation of the last (p − c) coordinates as well as under those of the first c coordinates are better than the minimax estimators of Eaten. The estimators considered outperform the maximum likelihood estimator (MLE) under the Stein′s loss function as well. The method involved here is computation of an unbiased estimate of the risk of an invariant estimator considered in this article. In addition we discuss its application to the problem of estimating a covariance matrix in a GMANOVA model since the estimation problem of the covariance matrix with extra data can be regarded as its canonical form.