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Generalized Bayes minimax estimators of the mean of multivariate normal distribution with unknown variance

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  • Wells, Martin T.
  • Zhou, Gongfu

Abstract

We construct a broad class of generalized Bayes minimax estimators of the mean of a multivariate normal distribution with covariance equal to [sigma]2Ip, with [sigma]2 unknown, and under the invariant loss ||[delta](X)-[theta]||2/[sigma]2. Examples that illustrate the theory are given. Most notably it is shown that a hierarchical version of the multivariate Student-t prior yields a Bayes minimax estimate.

Suggested Citation

  • Wells, Martin T. & Zhou, Gongfu, 2008. "Generalized Bayes minimax estimators of the mean of multivariate normal distribution with unknown variance," Journal of Multivariate Analysis, Elsevier, vol. 99(10), pages 2208-2220, November.
    • Handle: RePEc:eee:jmvana:v:99:y:2008:i:10:p:2208-2220
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    References listed on IDEAS

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    1. Maruyama Yuzo, 2003. "A robust generalized Bayes estimator improving on the James-Stein estimator for spherically symmetric distributions," Statistics & Risk Modeling, De Gruyter, vol. 21(1/2003), pages 69-78, January.
    2. Maruyama Yuzo, 1999. "Improving On The James-Stein Estimator," Statistics & Risk Modeling, De Gruyter, vol. 17(2), pages 137-140, February.
    3. Faith, Ray E., 1978. "Minimax Bayes estimators of a multivariate normal mean," Journal of Multivariate Analysis, Elsevier, vol. 8(3), pages 372-379, September.
    4. Fourdrinier, Dominique & Strawderman, William E. & Wells, Martin T., 2003. "Robust shrinkage estimation for elliptically symmetric distributions with unknown covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 85(1), pages 24-39, April.
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    Cited by:

    1. Zinodiny, S. & Rezaei, S. & Arjmand, O. Naghshineh & Nadarajah, S., 2013. "Bayes minimax estimation of the multivariate normal mean vector under quadratic loss functions," Statistics & Probability Letters, Elsevier, vol. 83(9), pages 2052-2056.
    2. Maruyama, Yuzo & Strawderman, William E., 2009. "An extended class of minimax generalized Bayes estimators of regression coefficients," Journal of Multivariate Analysis, Elsevier, vol. 100(10), pages 2155-2166, November.
    3. Zinodiny, S. & Strawderman, W.E. & Parsian, A., 2011. "Bayes minimax estimation of the multivariate normal mean vector for the case of common unknown variance," Journal of Multivariate Analysis, Elsevier, vol. 102(9), pages 1256-1262, October.
    4. Hisayuki Tsukuma & Tatsuya Kubokawa, 2014. "Estimation and Prediction Intervals in Transformed Linear Mixed Models," CIRJE F-Series CIRJE-F-930, CIRJE, Faculty of Economics, University of Tokyo.
    5. Shokofeh Zinodiny & Saralees Nadarajah, 2023. "Generalized Bayes Minimax Estimators of the Variance of a Multivariate Normal Distribution," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(2), pages 1667-1683, August.
    6. Tsukuma, Hisayuki & Kubokawa, Tatsuya, 2015. "Estimation of the mean vector in a singular multivariate normal distribution," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 245-258.
    7. Zinodiny, S. & Rezaei, S. & Nadarajah, S., 2014. "Bayes minimax estimation of the multivariate normal mean vector under balanced loss function," Statistics & Probability Letters, Elsevier, vol. 93(C), pages 96-101.

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