Minimax estimation of means of multivariate normal mixtures

Assume X = (X₁, ..., X_p)′ is a normal mixture distribution with density w.r.t. Lebesgue measure, f(x)=∫ 1 (2π)^p/2δ{divides}σ{divides}^1/2e ^{-(x-θ)′(σ-1(x-θ))/2σ 2} dF(σ), where Σ is a known positive definite matrix and F is any known c.d.f. on (0, ∞). Estimation of the mean vector under an arbitrary known quadratic loss function L_Q(θ, a) = (a - θ)′ Q(a - θ), Q a positive definite matrix, is considered. An unbiased estimator of risk is obatined for an arbitrary estimator, and a sufficient condition for estimators to be minimax is then achieved. The result is applied to modifying all the Stein estimators for the means of independent normal random variables to be minimax estimators for the problem considered here. In particular the results apply to the Stein class of limited translation estimators.