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A Revisit to Estimation of the Precision Matrix of the Wishart Distribution

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  • Tatsuya Kubokawa

    (Faculty of Economics, The University of Tokyo)

Abstract

The estimation of the precision matrix of the Wishart distribution is one of classical problems studied in a decision-theoretic framework and is related to estimation of mean and covariance matrices of a multivariate normal distribution. This paper revisits the estimation problem of the precision matrix and investigates how it connects with the theory of the covariance estimation from a decision-theoretic aspect. To evaluate estimators in terms of risk functions, we employ two kinds of loss functions: the non-scale-invariant loss and the scale-invariant loss functions which are induced from estimation of means. Using the same methods as in the estimation of the covariance matrix, we derive not only the James-Stein type of estimators improving on the Stein type one under the non-scale-invariant loss. It is observed that dominance properties given in the estimation of the covariance matrix do not necessarily hold in our setup under the non-scale-invariant loss, but still hold relative to the scale-invariant loss. The simulation studies are given, and estimators having superior risk performances are proposed.

Suggested Citation

  • Tatsuya Kubokawa, 2004. "A Revisit to Estimation of the Precision Matrix of the Wishart Distribution," CIRJE F-Series CIRJE-F-264, CIRJE, Faculty of Economics, University of Tokyo.
  • Handle: RePEc:tky:fseres:2004cf264
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    File URL: http://www.cirje.e.u-tokyo.ac.jp/research/dp/2004/2004cf264.pdf
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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. Zheng, Z., 1986. "On estimation of matrix of normal mean," Journal of Multivariate Analysis, Elsevier, vol. 18(1), pages 70-82, February.
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