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Estimating covariance matrices II

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  • Loh, Wei-Liem

Abstract

Let S1 and S2 be two independent p - p Wishart matrices with S1 ~ Wp([Sigma]1, n1) and S2 ~ Wp([Sigma]2, n2). We wish to estimate [zeta] = [Sigma]2[Sigma]1-1 under the loss function L1 = tr([zeta] - [zeta])' [Sigma]2-1([zeta] - [zeta]) [Sigma]1/tr [zeta]. By extending the techniques of Berger, Haff, and Stein for the one sample problem, alternative estimators to the usual estimators for [zeta] are obtained. However, the risks of these estimators are not available in closed form. A Monte Carlo study is used instead to evaluate their risk performances. The results indicate that the alternative estimators have excellent risk properties with respect to the usual estimators. In particular, dramatic savings in risk are obtained when the eigenvalues of [Sigma]2[Sigma]1-1 are close together.

Suggested Citation

  • Loh, Wei-Liem, 1991. "Estimating covariance matrices II," Journal of Multivariate Analysis, Elsevier, vol. 36(2), pages 163-174, February.
    • Handle: RePEc:eee:jmvana:v:36:y:1991:i:2:p:163-174
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    Citations

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    Cited by:

    1. 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.
    2. Brett Naul & Bala Rajaratnam & Dario Vincenzi, 2016. "The role of the isotonizing algorithm in Stein’s covariance matrix estimator," Computational Statistics, Springer, vol. 31(4), pages 1453-1476, December.
    3. Tsai, Ming-Tien & Kubokawa, Tatsuya, 2007. "Estimation of Wishart mean matrices under simple tree ordering," Journal of Multivariate Analysis, Elsevier, vol. 98(5), pages 945-959, May.
    4. Tsukuma, Hisayuki & Kubokawa, Tatsuya, 2007. "Methods for improvement in estimation of a normal mean matrix," Journal of Multivariate Analysis, Elsevier, vol. 98(8), pages 1592-1610, September.
    5. Wen, Jun, 2018. "Estimation of two high-dimensional covariance matrices and the spectrum of their ratio," Journal of Multivariate Analysis, Elsevier, vol. 168(C), pages 1-29.
    6. Konno, Yoshihiko, 2009. "Shrinkage estimators for large covariance matrices in multivariate real and complex normal distributions under an invariant quadratic loss," Journal of Multivariate Analysis, Elsevier, vol. 100(10), pages 2237-2253, November.
    7. Tsai, Ming-Tien, 2007. "Maximum likelihood estimation of Wishart mean matrices under Löwner order restrictions," Journal of Multivariate Analysis, Elsevier, vol. 98(5), pages 932-944, May.
    8. Perron, François, 1997. "On a Conjecture of Krishnamoorthy and Gupta, ," Journal of Multivariate Analysis, Elsevier, vol. 62(1), pages 110-120, July.
    9. 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.
    10. Hisayuki Tsukuma & Tatsuya Kubokawa, 2005. "Methods for Improvement in Estimation of a Normal Mean Matrix," CIRJE F-Series CIRJE-F-378, CIRJE, Faculty of Economics, University of Tokyo.

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