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On improved estimation of normal precision matrix and discriminant coefficients

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  • Tsukuma, Hisayuki
  • Konno, Yoshihiko
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    Abstract

    The problem of estimating the precision matrix of a multivariate normal distribution model is considered with respect to a quadratic loss function. A number of covariance estimators originally intended for a variety of loss functions are adapted so as to obtain alternative estimators of the precision matrix. It is shown that the alternative estimators have analytically smaller risks than the unbiased estimator of the precision matrix. Through numerical studies of risk values, it is shown that the new estimators have substantial reduction in risk. In addition, we consider the problem of the estimation of discriminant coefficients, which arises in linear discriminant analysis when Fisher's linear discriminant function is viewed as the posterior log-odds under the assumption that two classes differ in mean but have a common covariance matrix. The above method is also adapted for this problem in order to obtain improved estimators of the discriminant coefficients under the quadratic loss function. Furthermore, a numerical study is undertaken to compare the properties of a collection of alternatives to the "unbiased" estimator of the discriminant coefficients.

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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 97 (2006)
    Issue (Month): 7 (August)
    Pages: 1477-1500

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    Handle: RePEc:eee:jmvana:v:97:y:2006:i:7:p:1477-1500

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    Related research

    Keywords: Bayes and empirical Bayes estimation Discriminant coefficients Log-odds Precision matrix Wishart distribution;

    References

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    1. Dey, Dipak K., 1987. "Improved estimation of a multinormal precision matrix," Statistics & Probability Letters, Elsevier, vol. 6(2), pages 125-128, November.
    2. Haff, L. R., 1977. "Minimax estimators for a multinormal precision matrix," Journal of Multivariate Analysis, Elsevier, vol. 7(3), pages 374-385, September.
    3. Dey, Dipak K. & Srinivasan, C., 1991. "On estimation of discriminant coefficients," Statistics & Probability Letters, Elsevier, vol. 11(3), pages 189-193, March.
    4. Perron, F., 1992. "Minimax estimators of a covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 43(1), pages 16-28, October.
    5. Dey D. K. & Ghosh M. & Srinivasan C., 1990. "A New Class Of Improved Estimators Of A Multinormal Precision Matrix," Statistics & Risk Modeling, De Gruyter, vol. 8(2), pages 141-152, February.
    6. Haff, L. R., 1979. "An identity for the Wishart distribution with applications," Journal of Multivariate Analysis, Elsevier, vol. 9(4), pages 531-544, December.
    7. 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, vol. 53(4), pages 760-768, December.
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    Cited by:
    1. Bodnar, Taras & Okhrin, Yarema, 2008. "Properties of the singular, inverse and generalized inverse partitioned Wishart distributions," Journal of Multivariate Analysis, Elsevier, vol. 99(10), pages 2389-2405, November.

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