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Minimax estimators of a covariance matrix

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  • Perron, F.

Abstract

Let S: p - p have a nonsingular Wishart distribution with unknown matrix [Sigma] and n degrees of freedom, n >= p. For estimating [Sigma], a family of minimax estimators, with respect to the entropy loss, is presented. These estimators are of the form (S) = R[Phi](L) Rt, where R is orthogonal, L and [Phi] are diagonal, and RLRt = S. Conditions under which the components of [Phi] and L follow the same order relation are stated (i.e., writing L = diag((l1, ..., lp)t) and [Phi] = diag(([phi]1, ..., [phi]p)t) it is true that [phi]1 >= ... >= [phi]p if and only if l1 >= ... >= lp). Simulation results are included.

Suggested Citation

  • Perron, F., 1992. "Minimax estimators of a covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 43(1), pages 16-28, October.
    • Handle: RePEc:eee:jmvana:v:43:y:1992:i:1:p:16-28
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    Citations

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    Cited by:

    1. Konno, Yoshihiko, 2007. "Estimation of normal covariance matrices parametrized by irreducible symmetric cones under Stein's loss," Journal of Multivariate Analysis, Elsevier, vol. 98(2), pages 295-316, February.
    2. Tatsuya Kubokawa & M. S. Srivastava, 1999. ""Estimating the Covariance Matrix: A New Approach", June 1999," CIRJE F-Series CIRJE-F-52, CIRJE, Faculty of Economics, University of Tokyo.
    3. Kubokawa, T. & Srivastava, M. S., 2003. "Estimating the covariance matrix: a new approach," Journal of Multivariate Analysis, Elsevier, vol. 86(1), pages 28-47, July.
    4. 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.
    5. Tsukuma, Hisayuki & Kubokawa, Tatsuya, 2011. "Modifying estimators of ordered positive parameters under the Stein loss," Journal of Multivariate Analysis, Elsevier, vol. 102(1), pages 164-181, January.
    6. Konno, Yoshihiko, 2001. "Inadmissibility of the Maximum Likekihood Estimator of Normal Covariance Matrices with the Lattice Conditional Independence," Journal of Multivariate Analysis, Elsevier, vol. 79(1), pages 33-51, October.
    7. 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.
    8. Perron, François, 1997. "On a Conjecture of Krishnamoorthy and Gupta, ," Journal of Multivariate Analysis, Elsevier, vol. 62(1), pages 110-120, July.
    9. Tatsuya Kubokawa & M. S. Srivastava, 2002. "Estimating the Covariance Matrix: A New Approach," CIRJE F-Series CIRJE-F-162, CIRJE, Faculty of Economics, University of Tokyo.
    10. Besson, Olivier & Vincent, François & Gendre, Xavier, 2020. "A Stein’s approach to covariance matrix estimation using regularization of Cholesky factor and log-Cholesky metric," Statistics & Probability Letters, Elsevier, vol. 167(C).
    11. Hara, Hisayuki, 2001. "Other Classes of Minimax Estimators of Variance Covariance Matrix in Multivariate Normal Distribution," Journal of Multivariate Analysis, Elsevier, vol. 77(2), pages 175-186, May.
    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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