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Admissibility and minimaxity of Bayes estimators for a normal mean matrix

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  • Tsukuma, Hisayuki

Abstract

In some invariant estimation problems under a group, the Bayes estimator against an invariant prior has equivariance as well. This is useful notably for evaluating the frequentist risk of the Bayes estimator. This paper addresses the problem of estimating a matrix of means in normal distributions relative to quadratic loss. It is shown that a matricial shrinkage Bayes estimator against an orthogonally invariant hierarchical prior is admissible and minimax by means of equivariance. The analytical improvement upon every over-shrinkage equivariant estimator is also considered and this paper justifies the corresponding positive-part estimator preserving the order of the sample singular values.

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  • Tsukuma, Hisayuki, 2008. "Admissibility and minimaxity of Bayes estimators for a normal mean matrix," Journal of Multivariate Analysis, Elsevier, vol. 99(10), pages 2251-2264, November.
    • Handle: RePEc:eee:jmvana:v:99:y:2008:i:10:p:2251-2264
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    References listed on IDEAS

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    1. Sheena, Yo & Takemura, Akimichi, 1992. "Inadmissibility of non-order-preserving orthogonally invariant estimators of the covariance matrix in the case of Stein's loss," Journal of Multivariate Analysis, Elsevier, vol. 41(1), pages 117-131, April.
    2. Zheng, Z., 1986. "On estimation of matrix of normal mean," Journal of Multivariate Analysis, Elsevier, vol. 18(1), pages 70-82, February.
    3. Kubokawa, Tatsuya & Strawderman, William E., 2007. "On minimaxity and admissibility of hierarchical Bayes estimators," Journal of Multivariate Analysis, Elsevier, vol. 98(4), pages 829-851, April.
    4. Zheng, Z., 1986. "Selecting a minimax estimator doing well at a point," Journal of Multivariate Analysis, Elsevier, vol. 19(1), pages 14-23, June.
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    1. Tsukuma, Hisayuki, 2010. "Shrinkage priors for Bayesian estimation of the mean matrix in an elliptically contoured distribution," Journal of Multivariate Analysis, Elsevier, vol. 101(6), pages 1483-1492, July.
    2. Tsukuma, Hisayuki & Kubokawa, Tatsuya, 2017. "Proper Bayes and minimax predictive densities related to estimation of a normal mean matrix," Journal of Multivariate Analysis, Elsevier, vol. 159(C), pages 138-150.
    3. Zinodiny, S. & Rezaei, S. & Nadarajah, S., 2017. "Bayes minimax estimation of the mean matrix of matrix-variate normal distribution under balanced loss function," Statistics & Probability Letters, Elsevier, vol. 125(C), pages 110-120.
    4. Tsukuma, Hisayuki, 2009. "Generalized Bayes minimax estimation of the normal mean matrix with unknown covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 100(10), pages 2296-2304, November.
    5. Matsuda, Takeru & Strawderman, William E., 2019. "Improved loss estimation for a normal mean matrix," Journal of Multivariate Analysis, Elsevier, vol. 169(C), pages 300-311.
    6. Hu, Guikai & Peng, Ping, 2012. "Matrix linear minimax estimators in a general multivariate linear model under a balanced loss function," Journal of Multivariate Analysis, Elsevier, vol. 111(C), pages 286-295.
    7. Hoff, Peter D., 2011. "Hierarchical multilinear models for multiway data," Computational Statistics & Data Analysis, Elsevier, vol. 55(1), pages 530-543, January.
    8. Kubokawa Tatsuya & Tsukuma Hisayuki, 2009. "Minimaxity of the Stein risk-minimization estimator for a normal mean matrix," Statistics & Risk Modeling, De Gruyter, vol. 26(4), pages 243-261, July.

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