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Three Rank Formulas Associated With The Covariance Matrices Of The Blue And The Olse In The General Linear Model

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  • Puntanen, Simo
  • Styan, George P.H.
  • Tian, Yongge

Abstract

In this paper we consider the estimation of the expectation vector Xβ under the general linear model {y,Xβ,σ2V}. We introduce a new handy representation for the rank of the difference of the covariance matrices of the ordinary least squares estimator OLSE(Xβ) (= Hy, say) and the best linear unbiased estimator BLUE(Xβ) (= Gy, say). From this formula some well-known conditions for the equality between Hy and Gy follow at once. We recall that the equality between Hy and Gy can be characterized by the rank-subtractivity ordering between the covariance matrices of y and Hy. This rank characterization suggests a particular presentation for the rank of the difference of the covariance matrices of Hy and Gy. We show, however, that this presentation is valid if and only if the model is connected.

Suggested Citation

  • 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.
    • Handle: RePEc:cup:etheor:v:21:y:2005:i:03:p:659-663_05
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    Cited by:

    1. 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.
    2. Yuqin Sun & Rong Ke & Yongge Tian, 2014. "Some overall properties of seemingly unrelated regression models," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 98(2), pages 103-120, April.
    3. Tian, Yongge & Jiang, Bo, 2016. "Equalities for estimators of partial parameters under linear model with restrictions," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 299-313.
    4. Zhiping Xiong & Yingying Qin, 2014. "Extremal Ranks of Some Nonlinear Matrix Expressions with Applications," Journal of Optimization Theory and Applications, Springer, vol. 163(2), pages 595-613, November.
    5. Yongge Tian & Wenxing Guo, 2016. "On comparison of dispersion matrices of estimators under a constrained linear model," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 25(4), pages 623-649, November.
    6. 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.
    7. Ren, Xingwei, 2014. "On the equivalence of the BLUEs under a general linear model and its restricted and stochastically restricted models," Statistics & Probability Letters, Elsevier, vol. 90(C), pages 1-10.
    8. Huang, Yunying & Zheng, Bing, 2015. "The additive and block decompositions about the WLSEs of parametric functions for a multiple partitioned linear regression model," Journal of Multivariate Analysis, Elsevier, vol. 133(C), pages 123-135.

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