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Some equalities and inequalities for covariance matrices of estimators under linear model

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  • Y. Tian

    (Central University of Finance and Economics)

Abstract

Best linear unbiased estimators (BLUEs) of unknown parameters under linear models have minimum covariance matrices in the Löwner partial ordering among all linear unbiased estimators of the unknown parameters. Hence, BLUEs’ covariance matrices are usually used as a criterion to compare optimality with other types of estimator. During this work, people often need to establish certain equalities and inequalities for BLUEs’ covariance matrices, and use them in statistical inference of regression models. This paper aims at establishing some analytical formulas for calculating ranks and inertias of BLUEs’ covariance matrices under general linear model, and using these formulas in the comparison of covariance matrices of BLUEs with other types of estimator. This is in fact a mathematical work, and some new tools in matrix analysis are essentially utilized.

Suggested Citation

  • Y. Tian, 2017. "Some equalities and inequalities for covariance matrices of estimators under linear model," Statistical Papers, Springer, vol. 58(2), pages 467-484, June.
    • Handle: RePEc:spr:stpapr:v:58:y:2017:i:2:d:10.1007_s00362-015-0707-x
    DOI: 10.1007/s00362-015-0707-x
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    References listed on IDEAS

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    1. Yongge Tian & M. Beisiegel & E. Dagenais & C. Haines, 2008. "On the natural restrictions in the singular Gauss–Markov model," Statistical Papers, Springer, vol. 49(3), pages 553-564, July.
    2. Yongge Tian & Jieping Zhang, 2011. "Some equalities for estimations of partial coefficients under a general linear regression model," Statistical Papers, Springer, vol. 52(4), pages 911-920, November.
    3. Dong, Baomin & Guo, Wenxing & Tian, Yongge, 2014. "On relations between BLUEs under two transformed linear models," Journal of Multivariate Analysis, Elsevier, vol. 131(C), pages 279-292.
    4. Louis Guttman, 1944. "General theory and methods for matric factoring," Psychometrika, Springer;The Psychometric Society, vol. 9(1), pages 1-16, March.
    5. Rao, C. Radhakrishna, 1973. "Representations of best linear unbiased estimators in the Gauss-Markoff model with a singular dispersion matrix," Journal of Multivariate Analysis, Elsevier, vol. 3(3), pages 276-292, September.
    6. Puntanen, Simo & Styan, George P.H. & Tian, Yongge, 2005. "Three Rank Formulas Associated With The Covariance Matrices Of The Blue And The Olse In The General Linear Model," Econometric Theory, Cambridge University Press, vol. 21(3), pages 659-663, June.
    7. Lu, Changli & Gan, Shengjun & Tian, Yongge, 2015. "Some remarks on general linear model with new regressors," Statistics & Probability Letters, Elsevier, vol. 97(C), pages 16-24.
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    Cited by:

    1. Nesrin Güler & Melek Eriş Büyükkaya & Melike Yiğit, 2022. "Comparison of Covariance Matrices of Predictors in Seemingly Unrelated Regression Models," Indian Journal of Pure and Applied Mathematics, Springer, vol. 53(3), pages 801-809, September.
    2. Jiang, Bo & Tian, Yongge, 2017. "Rank/inertia approaches to weighted least-squares solutions of linear matrix equations," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 400-413.

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