An Improved Estimator of the Generalized Variance

Abstract

A multivariate extension is made of Stein's result (1964) on the estimation of the normal variance. Here the generalized variance $|\Sigma|$ is being estimated from a Wishart random matrix $S: p \times p \sim W(n, \Sigma)$ and an independent normal random matrix $X: p \times k \sim N(\xi, \Sigma \otimes 1_k)$ with $\xi$ unknown. The main result is that the minimax, best affine equivariant estimator $((n + 2 - p)!/(n + 2)!)|S|$ is dominated by $\min\{((n + 2 - p)!/(n + 2)!)|S|, ((n + k + 2 - p)!/(n + k + 2)!)|S + XX'|\}$. It is obtained by a variant of Stein's method which exploits zonal polynomials.