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Estimation of normal covariance matrices parametrized by irreducible symmetric cones under Stein's loss

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  • Konno, Yoshihiko

Abstract

In this paper the problem of estimating a covariance matrix parametrized by an irreducible symmetric cone in a decision-theoretic set-up is considered. By making use of some results developed in a theory of finite-dimensional Euclidean simple Jordan algebras, Bartlett's decomposition and an unbiased risk estimate formula for a general family of Wishart distributions on the irreducible symmetric cone are derived; these results lead to an extension of Stein's general technique for derivation of minimax estimators for a real normal covariance matrix. Specification of the results to the multivariate normal models with covariances which are parametrized by complex, quaternion, and Lorentz types gives minimax estimators for each model.

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  • 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.
    • Handle: RePEc:eee:jmvana:v:98:y:2007:i:2:p:295-316
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    1. Sheena, Yo & Takemura, Akimichi, 1992. "Inadmissibility of non-order-preserving orthogonally invariant estimators of the covariance matrix in the case of Stein's loss," Journal of Multivariate Analysis, Elsevier, vol. 41(1), pages 117-131, April.
    2. Andersson, Steen A. & Perlman, Michael D., 1984. "Two testing problems relating the real and complex multivariate normal distributions," Journal of Multivariate Analysis, Elsevier, vol. 15(1), pages 21-51, August.
    3. Shaman, Paul, 1980. "The inverted complex Wishart distribution and its application to spectral estimation," Journal of Multivariate Analysis, Elsevier, vol. 10(1), pages 51-59, March.
    4. 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.
    5. Perron, F., 1992. "Minimax estimators of a covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 43(1), pages 16-28, October.
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    1. 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.

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