MINQUE of Variance-Covariance Components in Linear Gauss-Markov Models
Publication: Journal of Surveying Engineering
Volume 137, Issue 4
Abstract
For heterogeneous and correlated observations, the variance components and the covariance components sometimes must be estimated. The forms of best invariant quadratic unbiased estimate (BIQUE) and Helmert-type estimation of variance and covariance components have already been derived by Koch and Grafarend, respectively. After obtaining the minimum norm quadratic unbiased estimate (MINQUE) of variance components, Rao derived only the MINQUE of the variance and covariance components for a special case in which the error vector is composed of a linear combination of independent random effect vectors of zero mean and the same variance-covariance matrix whose variance and covariance components were to be determined. However, an explicit expression of the MINQUE suitable to more general situations has not been derived. This paper defines the natural estimation of covariance components from errors and derives the MINQUE of variance and covariance components. The BIQUE and MINQUE of variance components without covariance components have the same iteration solution; the Helmert solution is only a special case of the MINQUE. However, the three estimates of variance and covariance components are different. The two MINQUE methods obtained in this paper have the advantage independence of the error distribution and offer a reasonable alternative in estimating variance and covariance components, and they can be used in the most general case. Numeric results show that the two MINQUE methods obtained in this paper are feasible.
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Acknowledgments
This project was sponsored by the “863” Project (2007AA12Z226) of the National Natural Science Foundation of China (NNSFCNSFC-41074009, NNSFCNSFC-40674015).
References
Amiri-Simkooei, A. R., Teunissen, P. J. G., and Tiberius, C. C. J. M. (2009). “Application of least-squares variance component estimation to GPS observation.” J. Surv. Eng., 135(4), 149–160.
Azais, J. M., Bardin, A., and Dhorne, T. (1993). “MINQE, maximum likelihood estimation and Fisher scoring algorithm for non linear variance models.” Stat., 24(3), 205–213.
Crocetto, N., Gatti, M., and Russo, P. (2000). “Simplified formulae for the BIQUE estimation of variance components in disjunctive observation groups.” J. Geodes., 74(6), 447–457.
Cui, X. Z. (2001). Generalized surveying adjustment, Wuhan University Press, Wuhan, P.R. China (in Chinese).
Drygas, H. (1977). “Best quadratic unbiased estimation in variance-covariance models.” Mathematische Operationsforschung und Statistik, Series Statistic, 8(2), 211–231.
Grafarend, E. W. (1980). “An introduction of variance-covariance-component estimation of Helmert type.” Zfv, 4, 161–180.
Grafarend, E. W. (1985). “Variance-covariance component estimation: Theoretical results and geodetic applications.” Stat. Decis., Supplemental Issue 2, 407–441.
Grafarend, E. W., and Schaffrin, B. (1976). “Equivalence of estimable quantities and invariance in geodetic networks.” Zfv, 101, 485–491.
Helmert, F. R. (1907). “Die ausgleichungsrechnung nach der methode der kleinsten quadrate.” ZweiteAuflage, Teubner, Leipzig, Germany.
Helmert, F. R. (1924). Die ausgleichsrechnung nach der methode der kleinsten quadrate, 3. Aufl., Teubner, Leipzig, Germany.
Henderson, C. R. (1953). “Estimation of variance and covariance components.” Biometrics, 9(2), 226–252.
Koch, K. R. (1986). “Maximum likelihood estimation of variance components.” Bull. Geodesique, 60(4), 329–338.
Koch, K. R. (1999). Parameter estimation and hypothesis testing in linear models, Springer, New York.
Kubik, K. (1970). “The estimation of the weights of measured quantities within the method of least squares.” Bull. Geodesique, 95(1), 21–40.
Lamotte, L. R. (1973). “Quadratic estimation of variance components.” Biometrics, 29, 311–330.
Malley, J. D. (1986). “Optimal unbiased estimation of variance components.” Lectures in statistics, Vol. 39, Springer-Verlag, Berlin.
Ou, Z. (1989). “Estimation of variance and covariance components.” Bull. Geodesique, 63, 139–148.
Peng, J. H. (2009). “Jointly robust estimation of unknown parameters and variance components based on expectation-maximization algorithm.” J. Surv. Eng., 135(1), 1–9.
Pukelsheim, F. (1974). “Schätzen von mittelwert und streuungmatrix in Gauss-Markov modellen.” Diplomarbeit, Univ. of Freiburg, Freiburg im Bresigau, Germany.
Rao, C. R. (1970). “Estimation of heteroscedastic variances in linear models.” J. Am. Stat. Assoc., 65(329), 161–171.
Rao, C. R. (1971a). “Estimation of variance and covariance components-MINQUE theory.” J. Multivariate Anal., 1(3), 257–275.
Rao, C. R. (1971b). “Minimum variance quadratic unbiased estimation of variance components.” J. Multivariate Anal., 1(4), 445–456.
Rao, C. R. (1972). “Estimation of variance and covariance components in linear models.” J. Am. Stat. Assoc., 67(337), 112–115.
Rao, C. R. (1973). Linear statistical inference and its application, Wiley, New York, 332–334.
Rao, C. R., and Kleffe, J. (1988). Estimation of variance components and applications, North-Holland, Amsterdam, Netherlands.
Rao, P. S. R. S. (1997). Variance components estimation, St. Edmundsbury, London.
Schaffrin, B. (1981a). “Ausgleichung mit bedingungs-ungleichungen.” AVN, 88, 227–238.
Schaffrin, B. (1981b). “Best invariant covariance component estimators and its application to the generalized multivariate adjustment of heterogeneous deformation observations.” Bull. Geodesique, 55(1), 73–85.
Schaffrin, B. (1983). Varianz-kovarianz-komponenten-schätzung bei der ausgleichung heterogener wiederholungsmessungen, C282, Deutsche Geodätische Kommission, Munich, Germany.
Searle, R. S. (1992). Variance components, Wiley, New York.
Seely, J. (1971). “Quadratic subspace and completeness.” Ann. Math. Stat., 42(2), 710–721.
Seely, J. (1972). “Completeness for a family of multivariate normal distributions.” Ann. Math. Stat., 43(5), 1644–1647.
Teunissen, P. J. G., and Amiri-Simkooei, A. R. (2007). “Least squares variance component estimation.” J. Geodes., 82(2), 65–82.
Verdooren, L. R. (1988). “Least squares estimators and non-negative estimators of variance components.” Commun. Stat., Theory Methods, 17(4), 1027–1051.
Volaufová, J. (1993). “MINQUE of variance components in replicated and multivariate linear model with linear restrictions.” QÜESTIIO, 17(2), 183–201.
Volaufová, J., and Witkovsky, V. (1992). “Estimation of variance components in mixed linear models.” Appl. Math., 37(2), 139–148.
Witkovsky, V. (1996). “On variance-covariance components estimation in linear models with AR (1) disturbances.” Acta Math. Univ. Comenianae, LXV(1), 129–139.
Xu, P., Liu, Y., Shen, Y., and Fukuda, Y. (2006). “Estimability analysis of variance and covariance components.” J. Geodes., 81(9), 593–602.
Yu, Z. C. (1992). “A generalization theory of estimation of variance-covariance components.” Manuscripta Geodaetica, 17, 295–301.
Yu, Z. C. (1996). “A universal formula of maximum likelihood estimation of variance-covariance components.” J. Geodes., 70(4), 233–240.
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Published In
Journal of Surveying Engineering
Volume 137 • Issue 4 • November 2011
Pages: 129 - 139
Copyright
© 2011 American Society of Civil Engineers.
History
Received: May 9, 2010
Accepted: Nov 24, 2010
Published online: Dec 9, 2010
Published in print: Nov 1, 2011
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