Minimax estimators for a multinormal precision matrix

Let Sp-p ~ Wishart ([Sigma], k), [Sigma] unknown, k > p + 1. Minimax estimators of [Sigma]-1 are given for L1, an Empirical Bayes loss function; and L2, a standard loss function (Ri [reverse not equivalent] E(Li | [Sigma]), I = 1, 2). The estimators are , a, b >= 0, r(·) a functional on Rp(p+2)/2. Stein, Efron, and Morris studied the special cases and , for certain, a, b. From their work , a = k - p - 1, b = p2 + p - 2; whereas, we prove . The reversal is surprising because a.e. (for a particular L2). Assume (compact) [subset of] , the set of p - p p.s.d. matrices. A "divergence theorem" on functions Fp-p : --> implies identities for Ri, i = 1, 2. Then, conditions are given for , i = 1, 2. Most of our results concern estimators with r(S) = t(U)/tr(S), U = p |S|1/p/tr(S).