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On estimation of a matrix of normal means with unknown covariance matrix

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  • Konno, Yoshihiko

Abstract

Let X be an m - p matrix normally distributed with matrix of means B and covariance matrix Im [circle times operator] [Sigma], where [Sigma] is a p - p unknown positive definite matrix. This paper studies the estimation of B relative to the invariant loss function tr . New classes of invariant minimax estimators are proposed for the case p > m + 1, which are multivariate extensions of the estimators of Stein and Baranchik. The method involves the unbiased estimation of the risk of an invariant estimator which depends on the eigenstructure of the usual F = XS-1Xt matrix, where S: p - p follows a Wishart matrix with n degrees of freedom and mean n[Sigma].

Suggested Citation

  • Konno, Yoshihiko, 1991. "On estimation of a matrix of normal means with unknown covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 36(1), pages 44-55, January.
  • Handle: RePEc:eee:jmvana:v:36:y:1991:i:1:p:44-55
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    References listed on IDEAS

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    1. Robinson, P M, 1987. "Asymptotically Efficient Estimation in the Presence of Heteroskedasticity of Unknown Form," Econometrica, Econometric Society, vol. 55(4), pages 875-891, July.
    2. Andrews, Donald W. K., 1988. "Chi-square diagnostic tests for econometric models : Introduction and applications," Journal of Econometrics, Elsevier, vol. 37(1), pages 135-156, January.
    3. Andrews, Donald W K, 1988. "Chi-Square Diagnostic Tests for Econometric Models: Theory," Econometrica, Econometric Society, vol. 56(6), pages 1419-1453, November.
    4. Pollard, David, 1985. "New Ways to Prove Central Limit Theorems," Econometric Theory, Cambridge University Press, vol. 1(03), pages 295-313, December.
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    Citations

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    Cited by:

    1. 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.
    2. Hisayuki Tsukuma & Tatsuya Kubokawa, 2014. "A Unified Approach to Estimating a Normal Mean Matrix in High and Low Dimensions," CIRJE F-Series CIRJE-F-926, CIRJE, Faculty of Economics, University of Tokyo.
    3. Tsukuma, Hisayuki, 2010. "Shrinkage minimax estimation and positive-part rule for a mean matrix in an elliptically contoured distribution," Statistics & Probability Letters, Elsevier, vol. 80(3-4), pages 215-220, February.
    4. Srivastava, M. S. & Kubokawa, T., 2005. "Minimax multivariate empirical Bayes estimators under multicollinearity," Journal of Multivariate Analysis, Elsevier, vol. 93(2), pages 394-416, April.
    5. Kubokawa, T. & Srivastava, M. S., 2002. "Estimating Risk and the Mean Squared Error Matrix in Stein Estimation," Journal of Multivariate Analysis, Elsevier, vol. 82(1), pages 39-64, July.
    6. Kubokawa, T. & Srivastava, M. S., 2001. "Robust Improvement in Estimation of a Mean Matrix in an Elliptically Contoured Distribution," Journal of Multivariate Analysis, Elsevier, vol. 76(1), pages 138-152, January.
    7. 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.
    8. Tatsuya Kubokawa & M. S. Srivastava, 2002. "Minimax Multivariate Empirical Bayes Estimators under Multicollinearity," CIRJE F-Series CIRJE-F-187, CIRJE, Faculty of Economics, University of Tokyo.
    9. Tsukuma, Hisayuki & Kubokawa, Tatsuya, 2007. "Methods for improvement in estimation of a normal mean matrix," Journal of Multivariate Analysis, Elsevier, vol. 98(8), pages 1592-1610, September.
    10. Hisayuki Tsukuma & Tatsuya Kubokawa, 2005. "Methods for Improvement in Estimation of a Normal Mean Matrix," CIRJE F-Series CIRJE-F-378, CIRJE, Faculty of Economics, University of Tokyo.
    11. Xu, Kai & He, Daojiang, 2015. "Further results on estimation of covariance matrix," Statistics & Probability Letters, Elsevier, vol. 101(C), pages 11-20.
    12. Oman, Samuel D., 2002. "Minimax Hierarchical Empirical Bayes Estimation in Multivariate Regression," Journal of Multivariate Analysis, Elsevier, vol. 80(2), pages 285-301, February.
    13. Tsukuma, Hisayuki & Kubokawa, Tatsuya, 2015. "A unified approach to estimating a normal mean matrix in high and low dimensions," Journal of Multivariate Analysis, Elsevier, vol. 139(C), pages 312-328.
    14. Tatsuka Kubokawa & M. S. Srivastava, 2002. "Prediction in Multivariate Mixed Linear Models," CIRJE F-Series CIRJE-F-180, CIRJE, Faculty of Economics, University of Tokyo.

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