ESTIMATION OF THE SCALE MATRIX AND ITS EIGENVALUES IN THE WISHART AND THE MULTIVARIATE F DISTRIBUTIONS

PUI LAM LEUNG AND WAI YIN CHAN

Department of Statistics, The Chinese University of Hong Kong, Shatin, Hong Kong

(Received June 12, 1996; revised April 25, 1997)

Abstract. In this paper, the problem of estimating the scale matrix and their eigenvalues in a Wishart distribution and in a multivariate F distribution (which arise naturally from a two-sample setting) are considered. A new class of estimators which shrink the eigenvalues towards their arithmetic mean are proposed. It is shown that the new estimator which dominates the usual unbiased estimator under the squared error loss function. A simulation study was carried out to study the performance of these estimators.

Key words and phrases: Covariance matrix, orthogonally invariant estimator, decision-theoretic estimation, shrinkage estimator.