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Estimations of parametric functions under a system of linear regression equations with correlated errors

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Abstract

Estimations of parametric functions under a system of linear regression equations with correlated errors across equations involve many complicated operations of matrices and their generalized inverses. In the past several years, a useful tool — the matrix rank method was utilized to simplify various complicated operations of matrices and their generalized inverses. In this paper, we use the matrix rank method to derive a variety of new algebraic and statistical properties for the best linear unbiased estimators (BLUEs) of parametric functions under the system. In particular, we give the necessary and sufficient conditions for some equalities, additive and block decompositions of BLUEs of parametric functions under the system to hold.

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References

  1. Alalouf, I. S., Styan, G. P. H.: Characterizations of estimability in the general linear model. Ann. Statist., 7, 194–200 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  2. Tian, Y., Beisiegel, M., Dagenais, E., et al.: On the natural restrictions in the singular Gauss-Markov model. Statist. Papers, 49, 553–564 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  3. Parks, R. W.: Efficient estimation of a system of regression equations when disturbances are both serially and contemporaneously correlated. J. Amer. Statist. Assoc., 62, 500–509 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  4. Rao, C. R.: Simultaneous estimation of parameters in different linear models and applications to biometric problems. Biometrics, 31, 545–554 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  5. Telser, L. G.: Iterative estimation of a set of linear regression equations. J. Amer. Statist. Assoc., 59, 845–862 (1964).

    Article  MATH  MathSciNet  Google Scholar 

  6. Groß, J.: Statistical estimation by a linear combination of two given statistics. Statist. Probab. Lett., 39, 379–384 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  7. Puntanen, S., Styan, G. P. H., Tian, Y.: Three rank formulas associated with the covariance matrices of the BLUE and the OLSE in the general linear model. Econometric Theory, 21, 659–664 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Qian, H., Tian, Y.: Partially superfluous observations. Econometric Theory, 22, 529–536 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Tian, Y.: Some decompositions of OLSEs and BLUEs under a partitioned linear model. Internat. Statist. Rev., 75, 224–248 (2007)

    Article  Google Scholar 

  10. Tian, Y.: On an additive decomposition of the BLUE in a multiple partitioned linear model. J. Multivariate Anal., 100, 767–776 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. Tian, Y.: On equalities for BLUEs under misspecified Gauss-Markov models. Acta Mathematica Sinica, English Series, 25(11), 1907–1920 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  12. Tian, Y., Puntanen, S.: On the equivalence of estimations under a general linear model and its transformed models. Linear Algebra Appl., 430, 2622–2641 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Tian, Y., Takane, Y.: Some algebraic and statistical properties of estimators under a general growth curve model. Electron. J. Linear Algebra, 6, 187–203 (2007)

    MathSciNet  Google Scholar 

  14. Tian, Y., Takane, Y.: On sum decompositions of weighted least-squares estimators under the partitioned linear model. Comm. Statist. Theory Methods, 37, 55–69 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Tian, Y., Takane, Y.: Some properties of projectors associated with the WLSE under a general linear model. J. Multivariate Anal., 99, 1070–1082 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  16. Tian, Y., Takane, Y.: On V-orthogonal projectors associated with a semi-norm. Ann. Inst. Statist. Math., 61, 517–530 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  17. Tian, Y., Takane, Y.: On consistency, natural restrictions and estimability under classical and extended growth curve models. J. Statist. Plann. Inference, 139, 2445–2458 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  18. Tian, Y., Wiens, D. P.: On equality and proportionality of ordinary least-squares, weighted least-squares and best linear unbiased estimators in the general linear model. Statist. Prob. Lett., 76, 1265–1272 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  19. Marsaglia, G., Styan, G. P. H.: Equalities and inequalities for ranks of matrices. Linear and Multilinear Algebra, 2, 269–292 (1974)

    Article  MathSciNet  Google Scholar 

  20. Rao, C. R.: Representations of best linear unbiased estimators in the Gauss-Markoff model with a singular dispersion matrix. J. Multivariate Anal., 3, 276–292 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  21. Zellner, A.: An efficiency method of estimating seemingly unrelated regression equations and tests for aggregation bias. J. Amer. Statist. Assoc., 57, 348–368 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  22. Zellner, A.: Estimators for seemingly unrelated regression equations: Some exact finite sample results. J. Amer. Statist. Assoc., 58, 977–992 (1963)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. China Economics and Management Academy, Central University of Finance and Economics, Beijing, 100081, P. R. China

    Yong Ge Tian

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  1. Yong Ge Tian
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Correspondence to Yong Ge Tian.

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Supported by National Natural Science Foundation of China (Grant No. 70871073)

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Tian, Y.G. Estimations of parametric functions under a system of linear regression equations with correlated errors. Acta. Math. Sin.-English Ser. 26, 1927–1942 (2010). https://doi.org/10.1007/s10114-010-8434-7

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  • DOI: https://doi.org/10.1007/s10114-010-8434-7

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