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A unified approach to estimating a normal mean matrix in high and low dimensions

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  • Tsukuma, Hisayuki
  • Kubokawa, Tatsuya

Abstract

This paper addresses the problem of estimating the normal mean matrix with an unknown covariance matrix. Motivated by an empirical Bayes method, we suggest a unified form of the Efron–Morris type estimators based on the Moore–Penrose inverse. This form not only can be defined for any dimension and any sample size, but also can contain the Efron–Morris type or Baranchik type estimators suggested so far in the literature. Also, the unified form suggests a general class of shrinkage estimators. For shrinkage estimators within the general class, a unified expression of unbiased estimators of the risk functions is derived regardless of the dimension of covariance matrix and the size of the mean matrix. An analytical dominance result is provided for a positive-part rule of the shrinkage estimators.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:jmvana:v:139:y:2015:i:c:p:312-328
    DOI: 10.1016/j.jmva.2015.04.003
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. Konno, Yoshihiko, 2009. "Shrinkage estimators for large covariance matrices in multivariate real and complex normal distributions under an invariant quadratic loss," Journal of Multivariate Analysis, Elsevier, vol. 100(10), pages 2237-2253, November.
    4. 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.
    5. 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.
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    Cited by:

    1. Tsukuma, Hisayuki & Kubokawa, Tatsuya, 2016. "Unified improvements in estimation of a normal covariance matrix in high and low dimensions," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 233-248.
    2. 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.
    3. Yuasa, Ryota & Kubokawa, Tatsuya, 2023. "Weighted shrinkage estimators of normal mean matrices and dominance properties," Journal of Multivariate Analysis, Elsevier, vol. 194(C).
    4. Tsukuma, Hisayuki, 2016. "Estimation of a high-dimensional covariance matrix with the Stein loss," Journal of Multivariate Analysis, Elsevier, vol. 148(C), pages 1-17.
    5. Fourdrinier, Dominique & Haddouche, Anis M. & Mezoued, Fatiha, 2021. "Covariance matrix estimation under data-based loss," Statistics & Probability Letters, Elsevier, vol. 177(C).
    6. Yuasa, Ryota & Kubokawa, Tatsuya, 2020. "Ridge-type linear shrinkage estimation of the mean matrix of a high-dimensional normal distribution," Journal of Multivariate Analysis, Elsevier, vol. 178(C).

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