Variance Component Estimation by the Method of Least-Squares

  • P.J.G. Teunissen
  • A.R. Amiri-Simkooei
Part of the International Association of Geodesy Symposia book series (IAG SYMPOSIA, volume 132)


Motivated by the fact that the method of least-squares is one of the leading principles in parameter estimation, we introduce and develop the method of least-squares variance component estimation (LS-VCE). The results are presented both for the model of observation equations and for the model of condition equations. LS-VCE has many attractive features. It provides a unified least-squares framework for estimating the unknown parameters of both the functional and stochastic model. Also, our existing body of knowledge of least-squares theory is directly applicable to LS-VCE. LS-VCE has a similar insightful geometric interpretation as standard least-squares. Properties of the normal equations, estimability, orthogonal projectors, precision of estimators, nonlinearity, and prior information on VCE can be easily established. Also measures of inconsistency, such as the quadratic form of residuals and the w-test statistic can directly be given. This will lead us to apply hypotheses testing to the stochastic model.


Least-squares variance component estimation BIQUE MINQUE REML 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Amiri-Simkooei, A. R. (2003). Formulation of L1 norm minimization in Gauss-Markov models. Journal of Surveying Engineering, 129(1), 37–43.CrossRefGoogle Scholar
  2. Amiri-Simkooei, A. R. (2007). Least-squares variance component estimation: theory and GPS applications. Ph.D. thesis, Delft University of Technology, Publication on Geodesy, 64, Netherlands Geodetic Commission, Delft.Google Scholar
  3. Caspary, W. F. (1987). Concepts of network and deformation analysis. Technical report, School of Surveying, The University of New South Wales, Kensington.Google Scholar
  4. Koch, K. R. (1978). Schätzung von varianzkomponenten. Allgemeine Vermessungs Nachrichten, 85, 264–269.Google Scholar
  5. Koch, K. R. (1986). Maximum likelihood estimate of variance components. Bulletin Geodesique, 60, 329–338. Ideas by A.J. Pope.CrossRefGoogle Scholar
  6. Koch, K. R. (1999). Parameter estimation and hypothesis testing in linear models. Springer Verlag, Berlin.Google Scholar
  7. Magnus, J. R. (1988). Linear Structures. Oxford University Press, London School of Economics and Political Science, Charles Griffin & Company LTD, London.Google Scholar
  8. Rao, C. R. (1971). Estimation of variance and covariance components - MINQUE theory. Journal of multivariate analysis, 1, 257–275.CrossRefGoogle Scholar
  9. Rao, C. R. and Kleffe, J. (1988). Estimation of variance components and applications, volume 3. North-Holland. Series in Statistics and Probability.Google Scholar
  10. Schaffrin, B. (1983). Varianz-kovarianz-komponenten-schätzung bei der ausgleichung heterogener wiederholungsmessungen. C282, Deutsche Geodätische Kommission, München.Google Scholar
  11. Sjöberg, L. E. (1983). Unbiased estimation of variance-covariance components in condition adjustment with unknowns – a MINQUE approach. Zeitschrift für Vermessungswesen, 108(9), 382–387.Google Scholar
  12. Teunissen, P. J. G. (1988). Towards a least-squares framework for adjusting and testing of both functional and stochastic model. Internal research memo, Geodetic Computing Centre, Delft. A reprint of original 1988 report is also available in 2004, No. 26, Scholar
  13. Teunissen, P. J. G. (1990). Nonlinear least-squares. Manuscripta Geodetica, 15(3), 137–150.Google Scholar
  14. Teunissen, P. J. G. and Amiri-Simkooei, A. R. (2008). Least-squares variance component estimation. Journal of Geodesy (in press), doi 10.1007/s00190-007-0157-x.Google Scholar
  15. Xu, P. L., Liu, Y. M., and Shen, Y. Z. (2007). Estimability analysis of variance and covariance components. Journal of Geodesy, 81(9), 593–602, doi 10.1007/s00190-006-0122-0.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • P.J.G. Teunissen
    • 1
  • A.R. Amiri-Simkooei
    • 1
  1. 1.Delft institute of Earth Observation and Space systems (DEOS), Delft University of Technology2629 HS DelftThe Netherlands

Personalised recommendations