Research Article Open Access

Risks Ratios of Shrinkage Estimators for the Multivariate Normal Mean

Abdenour Hamdaoui1 and Nadia Mezouar2
  • 1 University of Sciences and Technology Mohamed Boudiaf, Algeria
  • 2 Djillali Liabes University, Algeria

Abstract

We study the estimation of the mean θ of a multivariate Gaussian random variable XNp(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.

Journal of Mathematics and Statistics
Volume 13 No. 2, 2017, 77-87

DOI: https://doi.org/10.3844/jmssp.2017.77.87

Submitted On: 4 November 2016 Published On: 3 April 2017

How to Cite: Hamdaoui, A. & Mezouar, N. (2017). Risks Ratios of Shrinkage Estimators for the Multivariate Normal Mean. Journal of Mathematics and Statistics, 13(2), 77-87. https://doi.org/10.3844/jmssp.2017.77.87

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Keywords

  • James-Stein Estimator
  • Non-Central Chi-Square Distribution
  • Quadratic Risk
  • Shrinkage Estimator