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Representations of best linear unbiased estimators in the Gauss-Markoff model with a singular dispersion matrix

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  • Rao, C. Radhakrishna

Abstract

In the general Gauss-Markoff model (Y, X[beta], [sigma]2V), when V is singular, there exist linear functions of Y which vanish with probability 1 imposing some restrictions on Y as well as on the unknown [beta]. In all earlier work on linear estimation, representations of best-linear unbiased estimators (BLUE's) are obtained under the assumption: "L'Y is unbiased for X[beta] => L'X = X." Such a condition is not, however, necessary. The present paper provides all possible representations of the BLUE's some of which violate the condition L'X = X. Representations of X for given classes of BLUE's are also obtained.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:jmvana:v:3:y:1973:i:3:p:276-292
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    Cited by:

    1. 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.
    2. 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.
    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. Harry Haupt & Walter Oberhofer, 2002. "Fully restricted linear regression: A pedagogical note," Economics Bulletin, AccessEcon, vol. 3(1), pages 1-7.
    5. repec:spr:stpapr:v:58:y:2017:i:2:d:10.1007_s00362-015-0707-x is not listed on IDEAS
    6. 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.
    7. repec:ebl:ecbull:v:3:y:2002:i:1:p:1-7 is not listed on IDEAS
    8. Markiewicz, Augustyn, 1998. "Comparison of linear restricted models with respect to the validity of admissible and linearly sufficient estimators," Statistics & Probability Letters, Elsevier, vol. 38(4), pages 347-354, July.
    9. 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.
    10. Groß, Jürgen & Trenkler, Götz, 1998. "The equality between linear transforms of ordinary least squares and best linear unbiased estimator," Technical Reports 1998,14, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    11. Haupt, Harry & Oberhofer, Walter, 2006. "Best affine unbiased representations of the fully restricted general Gauss-Markov model," Journal of Multivariate Analysis, Elsevier, vol. 97(3), pages 759-764, March.
    12. 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.
    13. 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.
    14. repec:spr:stpapr:v:58:y:2017:i:2:d:10.1007_s00362-015-0706-y is not listed on IDEAS
    15. Gro[beta], Jürgen, 1998. "Statistical estimation by a linear combination of two given statistics," Statistics & Probability Letters, Elsevier, vol. 39(4), pages 379-384, August.
    16. He, Daojiang & Wu, Jie, 2014. "Admissible linear estimators of multivariate regression coefficient with respect to an inequality constraint under matrix balanced loss function," Journal of Multivariate Analysis, Elsevier, vol. 129(C), pages 37-43.
    17. 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.
    18. 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.
    19. repec:spr:stmapp:v:26:y:2017:i:4:d:10.1007_s10260-017-0381-3 is not listed on IDEAS
    20. Xuan Zhang & Yongge Tian, 2016. "On decompositions of BLUEs under a partitioned linear model with restrictions," Statistical Papers, Springer, vol. 57(2), pages 345-364, April.
    21. 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.
    22. 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.
    23. 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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