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Minimax estimation of a normal covariance matrix with the partial Iwasawa decomposition

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  • Tsukuma, Hisayuki

Abstract

This paper addresses the problem of estimating the normal covariance matrix relative to the Stein loss. The partial Iwasawa decomposition is used to reduce the original estimation problem to simultaneous estimation for variances and means of some normal distributions. The variances and the means are closely related to, respectively, the diagonal and the below-diagonal elements of a lower triangular matrix which is made from the Cholesky decomposition of the covariance matrix. Shrinkage type procedures are proposed for improvements not only on the diagonal elements but also on the below-diagonal elements corresponding to the James and Stein minimax estimator of the covariance matrix.

Suggested Citation

  • Tsukuma, Hisayuki, 2016. "Minimax estimation of a normal covariance matrix with the partial Iwasawa decomposition," Journal of Multivariate Analysis, Elsevier, vol. 145(C), pages 190-207.
    • Handle: RePEc:eee:jmvana:v:145:y:2016:i:c:p:190-207
    DOI: 10.1016/j.jmva.2015.12.013
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    References listed on IDEAS

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    1. Lee, Keunbaik & Yoo, Jae Keun, 2014. "Bayesian Cholesky factor models in random effects covariance matrix for generalized linear mixed models," Computational Statistics & Data Analysis, Elsevier, vol. 80(C), pages 111-116.
    2. 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.
    3. 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.
    4. Tsukuma, Hisayuki, 2014. "Minimax covariance estimation using commutator subgroup of lower triangular matrices," Journal of Multivariate Analysis, Elsevier, vol. 124(C), pages 333-344.
    5. Loh, Wei-Liem, 1991. "Estimating covariance matrices II," Journal of Multivariate Analysis, Elsevier, vol. 36(2), pages 163-174, February.
    6. Weiping Zhang & Chenlei Leng, 2012. "A moving average Cholesky factor model in covariance modelling for longitudinal data," Biometrika, Biometrika Trust, vol. 99(1), pages 141-150.
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