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Effect of adding regressors on the equality of the BLUEs under two linear models. (English) Zbl 1178.62068
Summary: We consider the estimation of regression coefficients in two partitioned linear models, shortly denoted as \({\mathcal M}_{12}=\{{\mathbf y},{\mathbf X}_1{\pmb\beta}_1+{\mathbf X}_2{\pmb\beta}_2,{\mathbf V}\}\), and \(\underline {{\mathcal M}}_{12}=\{{\mathbf y},{\mathbf X}_1{\pmb\beta}_1+{\mathbf X}_2{\pmb\beta}_2,\underline {{\mathbf V}}\}\), which differ only in their covariance matrices. We call \({\mathcal M}_{12}\) and \(\underline {{\mathcal M}}_{12}\) full models, and correspondingly, \({\mathcal M}_i=\{{\mathbf y},{\mathbf X}_i{\pmb\beta}_i,{\mathbf V}\}\) and \(\underline{\mathcal M}_i=\{{\mathbf y},{\mathbf X}_i{\pmb\beta}_i,{\mathbf V}\}\) small models. We give a necessary and sufficient condition for the equality between the best linear unbiased estimators (BLUEs) of \({\mathbf X}_1{\pmb\beta}_1\) under \({\mathcal M}_{12}\) and \(\underline{\mathcal M}_{12}\). In particular, we consider the equality of the BLUEs under the full models assuming that they are equal under the small models.

MSC:
62J05 Linear regression; mixed models
62H12 Estimation in multivariate analysis
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[1] Anderson, T.W., On the theory of testing serial correlation, Skandinavisk aktuarietidskrift, 31, 88-116, (1948) · Zbl 0033.08001
[2] Baksalary, J.K.; Mathew, T., Linear sufficiency and completeness in an incorrectly specified general gauss – markov model, Sankhyā, 48, 169-180, (1986) · Zbl 0611.62073
[3] Baksalary, J.K.; Mathew, T., Rank invariance criterion and its application to the unified theory of least squares, Linear algebra and its applications, 127, 393-401, (1990) · Zbl 0694.15003
[4] Chu, K.L.; Isotalo, J.; Puntanen, S.; Styan, G.P.H., On decomposing the Watson efficiency of ordinary least squares in a partitioned weakly singular linear model, Sankhyā, 66, 634-651, (2004) · Zbl 1193.62094
[5] Gross, J.; Puntanen, S., Estimation under a general partitioned linear model, Linear algebra and its applications, 321, 131-144, (2000) · Zbl 0966.62033
[6] Haslett, S.J., Updating linear models with dependent errors to include additional data and/or parameters, Linear algebra and its applications, 237/238, 329-349, (1996) · Zbl 0843.62072
[7] Isotalo, J.; Puntanen, S.; Styan, G.P.H., Effect of adding regressors on the equality of the OLSE and BLUE, International journal of statistical sciences, 6, 193-201, (2007)
[8] Isotalo, J.; Puntanen, S.; Styan, G.P.H., A useful matrix decomposition and its statistical applications in linear regression, Communications in statistics: theory and methods, 37, 1436-1457, (2008) · Zbl 1163.62051
[9] Mathew, T.; Bhimasankaram, P., On the robustness of LRT in singular linear models, Sankhyā, series A, 45, 301-312, (1983) · Zbl 0586.62043
[10] Marsaglia, G.; Styan, G.P.H., Equalities and inequalities for ranks of matrices, Linear and multilinear algebra, 2, 269-292, (1974)
[11] Mitra, S.K.; Moore, B.J., Gauss – markov estimation with an incorrect dispersion matrix, Sankhyā, series A, 35, 139-152, (1973) · Zbl 0277.62044
[12] Puntanen, S.; Styan, G.P.H., The equality of the ordinary least squares estimator and the best linear unbiased estimator [with comments by oscar kempthorne and by shayle R. searle and with “reply” by the authors], The American Statistician, 43, 153-164, (1989)
[13] Rao, C.R., Least squares theory using an estimated dispersion matrix and its application to measurement of signals, (), 355-372 · Zbl 0189.18503
[14] Rao, C.R., Representations of best linear unbiased estimators in the gauss – markov model with a singular dispersion matrix, Journal of multivariate analysis, 3, 276-292, (1973) · Zbl 0276.62068
[15] Rao, C.R.; Mitra, S.K., Generalized inverse of matrices and its applications, (1971), Wiley New York
[16] Sengupta, D.; Jammalamadaka, S.R., Linear models: an integrated approach, (2003), World Scientific River Edge, NJ · Zbl 1049.62080
[17] Werner, H.J.; Yapar, C., A BLUE decomposition in the general linear regression model, Linear algebra and its applications, 237/238, 395-404, (1996) · Zbl 0844.62061
[18] Zyskind, G., On canonical forms, non-negative covariance matrices and best and simple least squares linear estimators in linear models, The annals of mathematical statistics, 38, 1092-1109, (1967) · Zbl 0171.17103
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