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Inadmissibility of the Maximum Likekihood Estimator of Normal Covariance Matrices with the Lattice Conditional Independence

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  • Konno, Yoshihiko

Abstract

Lattice conditional independence (LCI) models introduced by S. A. Andersson and M. D. Perlman (1993, Ann. Statist.21, 1318-1358) have the pleasant feature of admitting explicit maximum likelihood estimators and likelihood ratio test statistics. This is because the likelihood function and parameter space for a LCI model can be factored into products of conditional likelihood functions and parameter spaces, where the standard multivariate techniques can be applied. In this paper we consider the problem of estimating the covariance matrices under LCI restriction in a decision theoretic setup. The Stein loss function is used in this study and, using the factorization mentioned above, minimax estimators are obtained. Since the maximum likelihood estimator has constant risk and is different from the minimax estimator, this shows that the maximum likelihood estimator under LCI restriction inadmissible. These results extend those obtained by W. James and C. Stein (1960, in "Proceedings of the Fourth Berkeley Symposium on Mathematics, Statistics, and Probability," Vol. 1, pp. 360-380, Univ. of California Press, Berkeley, CA) and D. K. Dey and C. Srinivasan (1985, Ann. Statist.13, 1581-1591) for estimating normal covariance matrices to the LCI models.

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  • 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.
    • Handle: RePEc:eee:jmvana:v:79:y:2001:i:1:p:33-51
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    References listed on IDEAS

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    1. D. R. Cox & Nanny Wermuth, 1999. "Likelihood Factorizations for Mixed Discrete and Continuous Variables," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 26(2), pages 209-220, June.
    2. 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.
    3. Konno, Y., 1995. "Estimation of a Normal Covariance Matrix with Incomplete Data under Stein's Loss," Journal of Multivariate Analysis, Elsevier, vol. 52(2), pages 308-324, February.
    4. Andersson, Steen A. & Perlman, Michael D., 1991. "Lattice-ordered conditional independence models for missing data," Statistics & Probability Letters, Elsevier, vol. 12(6), pages 465-486, December.
    5. Andersson, S. A. & Perlman, M. D., 1995. "Unbiasedness of the Likelihood Ratio Test for Lattice Conditional Independence Models," Journal of Multivariate Analysis, Elsevier, vol. 53(1), pages 1-17, April.
    6. 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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    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. Dongchu Sun & Xiaoqian Sun, 2005. "Estimation of the multivariate normal precision and covariance matrices in a star-shape model," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 57(3), pages 455-484, September.
    3. Sun, Xiaoqian & Sun, Dongchu, 2005. "Estimation of the Cholesky decomposition of the covariance matrix for a conditional independent normal model," Statistics & Probability Letters, Elsevier, vol. 73(1), pages 1-12, June.
    4. Sun, Dongchu & Sun, Xiaoqian, 2006. "Estimation of multivariate normal covariance and precision matrices in a star-shape model with missing data," Journal of Multivariate Analysis, Elsevier, vol. 97(3), pages 698-719, March.

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