Minimax estimators of a covariance matrix

Let S: p - p have a nonsingular Wishart distribution with unknown matrix [Sigma] and n degrees of freedom, n >= p. For estimating [Sigma], a family of minimax estimators, with respect to the entropy loss, is presented. These estimators are of the form (S) = R[Phi](L) Rt, where R is orthogonal, L and [Phi] are diagonal, and RLRt = S. Conditions under which the components of [Phi] and L follow the same order relation are stated (i.e., writing L = diag((l1, ..., lp)t) and [Phi] = diag(([phi]1, ..., [phi]p)t) it is true that [phi]1 >= ... >= [phi]p if and only if l1 >= ... >= lp). Simulation results are included.