Skip to main content
SpringerLink
Log in

On the weighted least-squares, the ordinary least-squares and the best linear unbiased estimators under a restricted growth curve model

  • Regular Article
  • Published:
Statistical Papers Aims and scope Submit manuscript

Abstract

Necessary and sufficient conditions are given for a restricted growth curve model to be consistent. The general expressions of the weighted least-squares estimators (WLSEs), the ordinary least-squares estimators (OLSEs) and the best linear unbiased estimator (BLUE) under this model are also derived. Moreover, some algebraic and statistical properties of these estimators are presented by rank method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Anna S (2012) Simultaneous choice of time points and the block design in the growth curve model Stat. Pap. doi:10.1007/s00362-012-0438-1

  • Baksalary JK, Puntanen S (1989) Weighted-least-squares estimation in the general Gauss–Markov model. In: Dodge Y (ed) Statistical data analysis and inference. Elsevier, Amsterdam, pp 355–368

  • Baksalary JK, Rao CR, Markiewicz A (1992) A study of the influence of the natural restrictions on estimation problems in the singular Gauss-Markov model. J Stat Plan Inference 31: 335–351

    Article  MATH  MathSciNet  Google Scholar 

  • Groß J (2004) The general Gauss–Markov model with possibly singular dispersion matrix. Stat Pap 45: 311–336

    Article  MATH  Google Scholar 

  • Khatri CG (1966) A note on a MANOVA model applied to problems in growth curve. Ann Inst Stat Math 18: 75–86

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Mitra SK (1984) The matrix equations AX=C,XB=D. Linear Algebra Appl 59: 171–181

    Article  MATH  MathSciNet  Google Scholar 

  • AB (1991) The matrix equation AXB + CYD = E over a principal ideal domain. SIAM J Matrix Anal Appl 12: 581–591

    Article  MATH  MathSciNet  Google Scholar 

  • Pan J, Fang KT (2002) Growth curve models with statistical diagnostics. Springer, New York

  • Potthoff R, Roy SN (1964) A generalized multivariate analysis of variance model useful especially for growth curve problems. Biometrika 51: 313–326

    MATH  MathSciNet  Google Scholar 

  • Rao CR (1965) The theory of least squares when the parameters are stochastic and ins application to the analysis of growth curves. Biometrika 52: 447–458

    MATH  MathSciNet  Google Scholar 

  • Rao CR (1966) Covariance adjustment and related problems in multivariate analysis. In: Krishnaiah PR (ed) Multivariate analysis. Academic Press, New York, pp 87–103

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

    Article  MATH  Google Scholar 

  • Seber GAF (1985) Multivariate observations. Wiley, New York

  • Tian Y, Takane Y (2007) Some algebraic and statistical properties of estimators under a general growth curve model. Electron J Linear Algebra 16: 187–203

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Tian Y, Beisiegel M, Dagenais E, Haines C (2008) On the natural restrictions in the singular Gauss–Markov model. Stat Pap 49: 553–564

    Article  MATH  MathSciNet  Google Scholar 

  • Woolson RF, Leeper JD (1980) Growth curve analysis of complete and incomplete longitudinal data. Commun Stat Theory Methods 9: 1491–1513

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Information Sciences, Weifang University, Weifang, 261061, People’s Republic of China

    Guang Jing Song

  2. Department of Mathematics, Shanghai University, Shanghai, 200444, People’s Republic of China

    Qing Wen Wang

Authors
  1. Guang Jing Song
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qing Wen Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guang Jing Song.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jing Song, G., Wen Wang, Q. On the weighted least-squares, the ordinary least-squares and the best linear unbiased estimators under a restricted growth curve model. Stat Papers 55, 375–392 (2014). https://doi.org/10.1007/s00362-012-0483-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00362-012-0483-9

Keywords

Mathematics Subject Classification

Search

Navigation