Let S1 and S2 be two independent p - p Wishart matrices with S1 ~ Wp([Sigma]1, n1) and S2 ~ Wp([Sigma]2, n2). We wish to estimate [zeta] = [Sigma]2[Sigma]1-1 under the loss function L1 = tr([zeta] - [zeta])' [Sigma]2-1([zeta] - [zeta]) [Sigma]1/tr [zeta]. By extending the techniques of Berger, Haff, and Stein for the one sample problem, alternative estimators to the usual estimators for [zeta] are obtained. However, the risks of these estimators are not available in closed form. A Monte Carlo study is used instead to evaluate their risk performances. The results indicate that the alternative estimators have excellent risk properties with respect to the usual estimators. In particular, dramatic savings in risk are obtained when the eigenvalues of [Sigma]2[Sigma]1-1 are close together.
