On improved estimators of the generalized variance

Treated in this paper is the problem of estimating with squared error loss the generalized variance [Sigma] from a Wishart random matrix S: p - p ~ Wp(n, [Sigma]) and an independent normal random matrix X: p - k ~ N([xi], [Sigma] [circle times operator] Ik) with [xi](p - k) unknown. Denote the columns of X by X(1) ,..., X(k) and set [psi](0)(S, X) = {(n - p + 2)!/(n + 2)!} S , [psi](i)(X, X) = min[[psi](i-1)(S, X), {(n - p + i + 2)!/(n + i + 2)!} S + X(1) X'(1) + ... + X(i) X'(i) ] and [Psi](i)(S, X) = min[[psi](0)(S, X), {(n - p + i + 2)!/(n + i + 2)!} S + X(1) X'(1) + ... + X(i) X'(i) ], i = 1,...,k. Our result is that the minimax, best affine equivariant estimator [psi](0)(S, X) is dominated by each of [Psi](i)(S, X), i = 1,...,k and for every i, [psi](i)(S, X) is better than [psi](i-1)(S, X). In particular, [psi](k)(S, X) = min[{(n - p + 2)!/(n + 2)!} S , {(n - p + 2)!/(n + 2)!} S + X(1)X'(1),..., {(n - p + k + 2)!/(n + k + 2)!} S + X(1)X'(1) + ... + X(k)X'(k)] dominates all other [psi]'s. It is obtained by considering a multivariate extension of Stein's result (Ann. Inst. Statist. Math. 16, 155-160 (1964)) on the estimation of the normal variance.