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Improved minimax estimation of the bivariate normal precision matrix under the squared loss

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  • Sun, Xiaoqian
  • Zhou, Xian

Abstract

Suppose that n independent observations are drawn from a multivariate normal distribution Np([mu],[Sigma]) with both mean vector [mu] and covariance matrix [Sigma] unknown. We consider the problem of estimating the precision matrix [Sigma]-1 under the squared loss . It is well known that the best lower triangular equivariant estimator of [Sigma]-1 is minimax. In this paper, by using the information in the sample mean on [Sigma]-1, we construct a new class of improved estimators over the best lower triangular equivariant minimax estimator of [Sigma]-1 for p=2. Our improved estimators are in the class of lower-triangular scale equivariant estimators and the method used is similar to that of Stein [1964. Inadmissibility of the usual estimator for the variance of a normal distribution with unknown mean. Ann. Inst. Statist. Math. 16, 155-160.]

Suggested Citation

  • Sun, Xiaoqian & Zhou, Xian, 2008. "Improved minimax estimation of the bivariate normal precision matrix under the squared loss," Statistics & Probability Letters, Elsevier, vol. 78(2), pages 127-134, February.
    • Handle: RePEc:eee:stapro:v:78:y:2008:i:2:p:127-134
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    References listed on IDEAS

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    1. Kubokawa, Tatsuya, 1989. "Improved estimation of a covariance matrix under quadratic loss," Statistics & Probability Letters, Elsevier, vol. 8(1), pages 69-71, May.
    2. Haff, L. R., 1979. "An identity for the Wishart distribution with applications," Journal of Multivariate Analysis, Elsevier, vol. 9(4), pages 531-544, December.
    3. Ghosh M. & Sinha B. K., 1987. "Inadmissibility Of The Best Equivariant Estimators Of The Variance-Covariance Matrix, The Precision Matrix, And The Generalized Variance Under Entropy Loss," Statistics & Risk Modeling, De Gruyter, vol. 5(3-4), pages 201-228, April.
    4. Xian Zhou & Xiaoqian Sun & Jinglong Wang, 2001. "Estimation of the Multivariate Normal Precision Matrix under the Entropy Loss," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(4), pages 760-768, December.
    5. Kubokawa, T. & Srivastava, M. S., 2003. "Estimating the covariance matrix: a new approach," Journal of Multivariate Analysis, Elsevier, vol. 86(1), pages 28-47, July.
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