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On improved estimation of normal precision matrix and discriminant coefficients

Author

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  • Tsukuma, Hisayuki
  • Konno, Yoshihiko

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.

Suggested Citation

  • Tsukuma, Hisayuki & Konno, Yoshihiko, 2006. "On improved estimation of normal precision matrix and discriminant coefficients," Journal of Multivariate Analysis, Elsevier, vol. 97(7), pages 1477-1500, August.
  • Handle: RePEc:eee:jmvana:v:97:y:2006:i:7:p:1477-1500
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    References listed on IDEAS

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    1. Dey, Dipak K. & Srinivasan, C., 1991. "On estimation of discriminant coefficients," Statistics & Probability Letters, Elsevier, vol. 11(3), pages 189-193, March.
    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. 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.
    4. Dey, Dipak K., 1987. "Improved estimation of a multinormal precision matrix," Statistics & Probability Letters, Elsevier, vol. 6(2), pages 125-128, November.
    5. 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.
    6. Haff, L. R., 1977. "Minimax estimators for a multinormal precision matrix," Journal of Multivariate Analysis, Elsevier, vol. 7(3), pages 374-385, September.
    7. Perron, F., 1992. "Minimax estimators of a covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 43(1), pages 16-28, October.
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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.
    2. Bodnar, Taras & Gupta, Arjun K. & Parolya, Nestor, 2016. "Direct shrinkage estimation of large dimensional precision matrix," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 223-236.
    3. Fourdrinier, Dominique & Mezoued, Fatiha & Wells, Martin T., 2016. "Estimation of the inverse scatter matrix of an elliptically symmetric distribution," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 32-55.

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