Risks Ratios of Shrinkage Estimators for the Multivariate Normal Mean

We study the estimation of the mean θ of a multivariate Gaussian random variable X∼N_p(θ,σ²I_p) in ℜ^p, σ² is unknown and estimated by the chi-square variable S²∼σ²χ_n². In this work we are interested in studying bounds and limits of risk ratios of shrinkage estimators to the maximum likelihood estimator X, when n and p tend to infinity. We recall that the risk ratios of shrinkage estimators to the maximum likelihood estimator has a lower bound B_m, when n and p tend to infinity. We show that if the shrinkage function ψ(S²,||X²||) satisfies some conditions, the risk ratios of shrinkage estimators (1-ψ(S²,||X²||)S²/||X²||)X, which did not inevitably minimax, to attain the limiting lower bound B_m which is strictly lower than 1.