IDEAS home Printed from https://ideas.repec.org/p/tky/fseres/2014cf926.html
   My bibliography  Save this paper

A Unified Approach to Estimating a Normal Mean Matrix in High and Low Dimensions

Author

Listed:
  • Hisayuki Tsukuma

    (Faculty of Medicine, Toho University)

  • Tatsuya Kubokawa

    (Faculty of Economics, The University of Tokyo)

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 uni ed form of the Efron-Morris type estimators based on the Moore-Penrose inverse. This form not only can be de ned 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 uni ed form suggests a general class of shrinkage estimators. For shrinkage estimators within the general class, a uni ed 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

  • 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.
    • Handle: RePEc:tky:fseres:2014cf926
    as

    Download full text from publisher

    File URL: http://www.cirje.e.u-tokyo.ac.jp/research/dp/2014/2014cf926.pdf
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    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. Bilodeau, Martin & Kariya, Takeaki, 1989. "Minimax estimators in the normal MANOVA model," Journal of Multivariate Analysis, Elsevier, vol. 28(2), pages 260-270, February.
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. Yuasa, Ryota & Kubokawa, Tatsuya, 2023. "Weighted shrinkage estimators of normal mean matrices and dominance properties," Journal of Multivariate Analysis, Elsevier, vol. 194(C).
    3. Xu, Kai & He, Daojiang, 2015. "Further results on estimation of covariance matrix," Statistics & Probability Letters, Elsevier, vol. 101(C), pages 11-20.
    4. 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).
    5. 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.
    6. 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.
    7. Shokofeh Zinodiny & Saralees Nadarajah, 2024. "A New Class of Bayes Minimax Estimators of the Mean Matrix of a Matrix Variate Normal Distribution," Mathematics, MDPI, vol. 12(7), pages 1-14, April.
    8. 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.
    9. 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.
    10. 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.
    11. 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.
    12. 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.
    13. 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.
    14. 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.
    15. Oman, Samuel D., 2002. "Minimax Hierarchical Empirical Bayes Estimation in Multivariate Regression," Journal of Multivariate Analysis, Elsevier, vol. 80(2), pages 285-301, February.
    16. Hisayuki Tsukuma & Tatsuya Kubokawa, 2014. "Unified Improvements in Estimation of a Normal Covariance Matrix in High and Low Dimesions," CIRJE F-Series CIRJE-F-937, CIRJE, Faculty of Economics, University of Tokyo.
    17. T Matsuda & W E Strawderman, 2022. "Estimation under matrix quadratic loss and matrix superharmonicity [Shrinkage estimation with a matrix loss function]," Biometrika, Biometrika Trust, vol. 109(2), pages 503-519.
    18. Tatsuya Kubokawa & Muni S. Srivastava, 2013. "Optimal Ridge-type Estimators of Covariance Matrix in High Dimension," CIRJE F-Series CIRJE-F-906, CIRJE, Faculty of Economics, University of Tokyo.
    19. Piotr Graczyk & Hideyuki Ishi & Salha Mamane, 2019. "Wishart exponential families on cones related to tridiagonal matrices," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(2), pages 439-471, April.
    20. Nhat Minh Nguyen & Trung Duc Nguyen & Eleftherios I. Thalassinos & Hoang Anh Le, 2022. "The Performance of Shrinkage Estimator for Stock Portfolio Selection in Case of High Dimensionality," JRFM, MDPI, vol. 15(6), pages 1-12, June.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:tky:fseres:2014cf926. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: CIRJE administrative office (email available below). General contact details of provider: https://edirc.repec.org/data/ritokjp.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.