IDEAS home Printed from https://ideas.repec.org/a/spr/stmapp/v25y2016i4d10.1007_s10260-016-0350-2.html
   My bibliography  Save this article

On comparison of dispersion matrices of estimators under a constrained linear model

Author

Listed:
  • Yongge Tian

    (Central University of Finance and Economics)

  • Wenxing Guo

    (Central University of Finance and Economics)

Abstract

We introduce some new mathematical tools in the analysis of dispersion matrices of the two well-known OLSEs and BLUEs under general linear models with parameter restrictions. We first establish some formulas for calculating the ranks and inertias of the differences of OLSEs’ and BLUEs’ dispersion matrices of parametric functions under the general linear model $${\mathscr {M}}= \{\mathbf{y}, \ \mathbf{X }\pmb {\beta }, \ \pmb {\Sigma }\}$$ M = { y , X β , Σ } and the constrained model $${\mathscr {M}}_r = \{\mathbf{y}, \, \mathbf{X }\pmb {\beta }\, | \, \mathbf{A }\pmb {\beta }= \mathbf{b}, \ \pmb {\Sigma }\}$$ M r = { y , X β | A β = b , Σ } , where $$\mathbf{A }\pmb {\beta }= \mathbf{b}$$ A β = b is a consistent linear matrix equation for the unknown parameter vector $$\pmb {\beta }$$ β to satisfy. As applications, we derive necessary and sufficient conditions for many equalities and inequalities of OLSEs’ and BLUEs’ dispersion matrices to hold under $${\mathscr {M}}$$ M and $${\mathscr {M}}_r$$ M r .

Suggested Citation

  • 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.
    • Handle: RePEc:spr:stmapp:v:25:y:2016:i:4:d:10.1007_s10260-016-0350-2
    DOI: 10.1007/s10260-016-0350-2
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10260-016-0350-2
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10260-016-0350-2?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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. Stephen Haslett & Simo Puntanen, 2011. "On the equality of the BLUPs under two linear mixed models," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 74(3), pages 381-395, November.
    3. Yongge Tian, 2010. "On equalities of estimations of parametric functions under a general linear model and its restricted models," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 72(3), pages 313-330, November.
    4. 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.
    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. Stephen Haslett & Jarkko Isotalo & Yonghui Liu & Simo Puntanen, 2014. "Equalities between OLSE, BLUE and BLUP in the linear model," Statistical Papers, Springer, vol. 55(2), pages 543-561, May.
    8. Stephen Haslett & Simo Puntanen, 2010. "Equality of BLUEs or BLUPs under two linear models using stochastic restrictions," Statistical Papers, Springer, vol. 51(2), pages 465-475, June.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. 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.
    3. Tian, Yongge & Zhang, Xuan, 2016. "On connections among OLSEs and BLUEs of whole and partial parameters under a general linear model," Statistics & Probability Letters, Elsevier, vol. 112(C), pages 105-112.
    4. 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.
    5. 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.
    6. 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.
    7. S. J. Haslett & X. Q. Liu & A. Markiewicz & S. Puntanen, 2020. "Some properties of linear sufficiency and the BLUPs in the linear mixed model," Statistical Papers, Springer, vol. 61(1), pages 385-401, February.
    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.
    9. Yongge Tian, 2017. "Transformation approaches of linear random-effects models," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 26(4), pages 583-608, November.
    10. Bo Jiang & Yuqin Sun, 2019. "On the equality of estimators under a general partitioned linear model with parameter restrictions," Statistical Papers, Springer, vol. 60(1), pages 273-292, February.
    11. Liu, Xin & Wang, Qing-Wen, 2013. "Equality of the BLUPs under the mixed linear model when random components and errors are correlated," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 297-309.
    12. Ren, Xingwei, 2016. "Estimation in singular linear models with stepwise inclusion of linear restrictions," Journal of Multivariate Analysis, Elsevier, vol. 148(C), pages 60-72.
    13. 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.
    14. Stephen Haslett & Simo Puntanen, 2011. "On the equality of the BLUPs under two linear mixed models," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 74(3), pages 381-395, November.
    15. Guang Jing Song & Qing Wen Wang, 2014. "On the weighted least-squares, the ordinary least-squares and the best linear unbiased estimators under a restricted growth curve model," Statistical Papers, Springer, vol. 55(2), pages 375-392, May.
    16. Changli Lu & Yuqin Sun & Yongge Tian, 2013. "On relations between weighted least-squares estimators of parametric functions under a general partitioned linear model and its small models," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 76(5), pages 707-722, July.
    17. B. Arendacká & S. Puntanen, 2015. "Further remarks on the connection between fixed linear model and mixed linear model," Statistical Papers, Springer, vol. 56(4), pages 1235-1247, November.
    18. Changli Lu & Yuqin Sun & Yongge Tian, 2018. "Two competing linear random-effects models and their connections," Statistical Papers, Springer, vol. 59(3), pages 1101-1115, September.
    19. 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.
    20. 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.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:stmapp:v:25:y:2016:i:4:d:10.1007_s10260-016-0350-2. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.