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Minimax estimators for a multinormal precision matrix

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  • Haff, L. R.

Abstract

Let Sp-p ~ Wishart ([Sigma], k), [Sigma] unknown, k > p + 1. Minimax estimators of [Sigma]-1 are given for L1, an Empirical Bayes loss function; and L2, a standard loss function (Ri [reverse not equivalent] E(Li | [Sigma]), I = 1, 2). The estimators are , a, b >= 0, r(·) a functional on Rp(p+2)/2. Stein, Efron, and Morris studied the special cases and , for certain, a, b. From their work , a = k - p - 1, b = p2 + p - 2; whereas, we prove . The reversal is surprising because a.e. (for a particular L2). Assume (compact) [subset of] , the set of p - p p.s.d. matrices. A "divergence theorem" on functions Fp-p : --> implies identities for Ri, i = 1, 2. Then, conditions are given for , i = 1, 2. Most of our results concern estimators with r(S) = t(U)/tr(S), U = p |S|1/p/tr(S).

Suggested Citation

  • Haff, L. R., 1977. "Minimax estimators for a multinormal precision matrix," Journal of Multivariate Analysis, Elsevier, vol. 7(3), pages 374-385, September.
    • Handle: RePEc:eee:jmvana:v:7:y:1977:i:3:p:374-385
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    Citations

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    Cited by:

    1. Yeil Kwon & Zhigen Zhao, 2023. "On F-modelling-based empirical Bayes estimation of variances," Biometrika, Biometrika Trust, vol. 110(1), pages 69-81.
    2. Joel Bun & Jean-Philippe Bouchaud & Marc Potters, 2016. "Cleaning large correlation matrices: tools from random matrix theory," Papers 1610.08104, arXiv.org.
    3. Kourtis, Apostolos & Dotsis, George & Markellos, Raphael N., 2012. "Parameter uncertainty in portfolio selection: Shrinking the inverse covariance matrix," Journal of Banking & Finance, Elsevier, vol. 36(9), pages 2522-2531.
    4. A. Grieve, 1988. "A further note on some wishart expectations," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 35(1), pages 197-202, December.
    5. Tsukuma, Hisayuki, 2014. "Bayesian estimation of a bounded precision matrix," Journal of Multivariate Analysis, Elsevier, vol. 127(C), pages 160-172.
    6. Pensky, Marianna, 1999. "Nonparametric Empirical Bayes Estimation of the Matrix Parameter of the Wishart Distribution," Journal of Multivariate Analysis, Elsevier, vol. 69(2), pages 242-260, May.
    7. Sheena Yo & Gupta Arjun K., 2003. "Estimation of the multivariate normal covariance matrix under some restrictions," Statistics & Risk Modeling, De Gruyter, vol. 21(4), pages 327-342, April.
    8. Chételat, Didier & Wells, Martin T., 2016. "Improved second order estimation in the singular multivariate normal model," Journal of Multivariate Analysis, Elsevier, vol. 147(C), pages 1-19.
    9. Xiangyu Cui & Xuan Zhang, 2021. "Index tracking strategy based on mixed-frequency financial data," PLOS ONE, Public Library of Science, vol. 16(4), pages 1-15, April.
    10. Hisayuki Tsukuma, 2003. "On estimation in multivariate linear calibration with elliptical errors," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 55(3), pages 447-466, September.
    11. Fourdrinier, Dominique & Mezoued, Fatiha & Wells, Martin T., 2016. "Estimation of the inverse scatter matrix of an elliptically symmetric distribution," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 32-55.
    12. Tsukuma, Hisayuki & Konno, Yoshihiko, 2006. "On improved estimation of normal precision matrix and discriminant coefficients," Journal of Multivariate Analysis, Elsevier, vol. 97(7), pages 1477-1500, August.

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