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On equalities of estimations of parametric functions under a general linear model and its restricted models

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Abstract

Estimations of parametric functions under a general linear model and its restricted models involve some complicated operations of matrices and their generalized inverses. In the past several years, a powerful tool—the matrix rank method was utilized to manipulate various complicated matrix expressions that involve generalized inverses of matrices. In this paper, we use this method to derive necessary and sufficient conditions for six equalities of the ordinary least-squares estimators and the best linear unbiased estimators of parametric functions to equal under a general linear model and its corresponding restricted model.

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References

  • Alalouf IS, Styan GPH (1979) Characterizations of estimability in the general linear model. Ann Stat 7: 194–200

    Article  MATH  MathSciNet  Google Scholar 

  • Amemiya T (1985) Advanced econometrics. Basil Blackwell, Oxford

    Google Scholar 

  • Baksalary JK, Kala R (1979) Best linear unbiased estimation in the restricted general linear model. Math Operationsforsch Stat Ser Stat 10: 27–35

    MATH  MathSciNet  Google Scholar 

  • Chipman JS, Rao MM (1964) The treatment of linear restrictions in regression analysis. Econometrica 32: 198–209

    Article  MATH  MathSciNet  Google Scholar 

  • Dent WT (1980) On restricted estimation in linear models. J Econom 12: 45–58

    MathSciNet  Google Scholar 

  • Drygas H (1970) The coordinate-free approach to Gauss-Markov estimation. Springer, Heidelberg

    MATH  Google Scholar 

  • Fomby T, Hill C, Johnson S (1984) Advanced econometric methods. Springer, New York

    MATH  Google Scholar 

  • Groß J, Trenkler G (1998) On the equality linear statistics in General Markov model. In: Mukherjee SP, Basu SK, Sinha BK (eds) Frontiers of Statistics. Narosa Publishing House, New Delhi, , pp 189–194

    Google Scholar 

  • Groß J, Trenkler G, Werner HJ (2001) The equality of linear transformations of the ordinary least squares estimator and the best linear unbiased estimator. Sankhyā Ser A 63: 118–127

    MATH  Google Scholar 

  • Hallum CR, Lewis TO, Boullion TL (1973) Estimation in the restricted general linear model with a positive semidefinite covariance matrix. Comm Stat 1: 157–166

    Article  MathSciNet  Google Scholar 

  • Harville DA (2008) Matrix algebra from a statistician’s perspective. Springer, New York

    Book  MATH  Google Scholar 

  • Haupt H, Oberhofer W (2002) Fully restricted linear regression: a pedagogical note. Econ Bull 3: 1–7

    Google Scholar 

  • Isotalo J, Puntanen S (2009) A note on the equality of the OLSE and the BLUE of the parametric functions in the general Gauss-Markov model. Stat Papers 50: 185–193

    Article  MATH  MathSciNet  Google Scholar 

  • Kruskal W (1968) When are gauss-Markov and least squares estimators identical? A coordinate free approach. Ann Math Stat 39: 70–75

    Article  MATH  MathSciNet  Google Scholar 

  • Kurata H (1998) A generalization of Rao’s covariance structure with applications to several linear models. J Multivar Anal 67: 297–305

    Article  MATH  MathSciNet  Google Scholar 

  • Magnus JR, Neudecker H (1999) Matrix differential calculus with applications in statistics and econometrics, 2nd edn. Wiley, New York

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Mathew T (1983) A note on best linear unbiased estimation in the restricted general linear model. Statistics 14: 3–6

    Article  MATH  MathSciNet  Google Scholar 

  • Penrose R (1955) A generalized inverse for matrices. Proc Cambridge Philos Soc 51: 406–413

    Article  MATH  MathSciNet  Google Scholar 

  • Puntanen S, Styan GPH (1989) The equality of the ordinary least squares estimator and the best linear unbiased estimator.With comments by Kempthorne O, Searle SR, and a reply by the authors. Am Stat 43:153–164

    Google Scholar 

  • Rao CR (1971) Unified theory of linear estimation. Sankhyā Ser A 33: 371–394

    MATH  Google Scholar 

  • Rao CR (1973) Representations of best linear unbiased estimators in the Gauss-Markoff model with a singular dispersion matrix. J Multivar Anal 3: 276–292

    Article  MATH  Google Scholar 

  • Rao CR, Rao MB (1998) Matrix algebra and its applications to statistics and econometrics. World Scientific, New York

    MATH  Google Scholar 

  • Ravikumar B, Ray S, Savin NE (2000) Robust wald tests in SUR systems with adding up restrictions. Econometrica 68: 715–719

    Article  MATH  MathSciNet  Google Scholar 

  • Roth WE (1952) The equations AX − YB = C and AX − XB = C in matrices. Proc Am Math Soc 3: 392–396

    MATH  Google Scholar 

  • Searle SR (2006) Matrix algebra useful for statistics. Wiley, New York

    MATH  Google Scholar 

  • Seber GAF (2008) A matrix handbook for statisticians. Wiley, New York

    Google Scholar 

  • Yang W-L, Cui H-J, Sun G-W (1987) On best linear unbiased estimation in the restricted general linear model. Statistics 18: 17–20

    Article  MATH  MathSciNet  Google Scholar 

  • Zyskind G (1967) On canonical forms, nonnegative covariance matrices, and best and simple least squares estimators in linear models. Ann Stat 38: 1092–1110

    Article  MATH  MathSciNet  Google Scholar 

Download references

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

    Yongge Tian

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  1. Yongge Tian
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Correspondence to Yongge Tian.

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Tian, Y. On equalities of estimations of parametric functions under a general linear model and its restricted models. Metrika 72, 313–330 (2010). https://doi.org/10.1007/s00184-009-0255-2

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  • DOI: https://doi.org/10.1007/s00184-009-0255-2

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