Monotonic minimax estimators of a 2×2 covariance matrix

Let S : 2 × 2 have a nonsingular Wishart distribution with unknown matrix σ and n degrees of freedom. For estimating σ two families of mimmax estimators, with respect to the entropy loss, are presented. These estimators are of the form σ(S) = Rø(L)R^t where R is orthogonal, L and Φ are diagonal, and RLR^T = S. Conditions under which the components of Φ and L follow the same order relation i.e., writing Φ = diag(Φ₁,Φ₂) and L = diag(l₁,/₂) with l₁ ≥ l₂, we have Φ₁ ≥ Φ₂] are established. Comparisons with Stein's estimators and other orthogonally invariant estimators are discussed.