Risks Ratios of Shrinkage Estimators for the Multivariate Normal Mean
Abdenour Hamdaoui and Nadia Mezouar
DOI : 10.3844/jmssp.2017.77.87
Journal of Mathematics and Statistics
Volume 13, Issue 2
We study the estimation of the mean θ of a multivariate Gaussian random variable X∼Np(θ,σ2Ip) in ℜp, σ2 is unknown and estimated by the chi-square variable S2∼σ2χn2. 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 Bm, 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 Bm which is strictly lower than 1.
© 2017 Abdenour Hamdaoui and Nadia Mezouar. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.