zbMATH — the first resource for mathematics

On decompositions of BLUEs under a partitioned linear model with restrictions. (English) Zbl 1341.62142
Summary: Estimators of parametric functions under a general linear regression model with restrictions involve lots of complicated operations of matrices and their generalized inverses. In this paper, we study relationships between the restricted partitioned linear model \(\{\mathbf{y}, \, \mathbf{X}_1 \boldsymbol{\beta}_1 + \mathbf{X}_2 \boldsymbol{\beta}_2 \mid \mathbf{A}_1\boldsymbol{\beta}_1=\mathbf{b}_1,\mathbf{A}_2\boldsymbol{\beta}_2=\mathbf{b}_2, \, \boldsymbol{\Sigma}\}\) and the corresponding two small restricted linear models \(\{ \mathbf{y}, \, \mathbf{X}_1 \boldsymbol{\beta}_1 \mid \mathbf{A}_1\boldsymbol{\beta}_1=\mathbf{b}_1, \, \boldsymbol{\Sigma}\}\) and \(\{\mathbf{y}, \, \mathbf{X}_2 \boldsymbol{\beta}_2 \mid \mathbf{A}_2\boldsymbol{\beta}_2=\mathbf{b}_2, \, \boldsymbol{\Sigma}\}\) by using various matrix rank formulas. In particular, we derive necessary and sufficient conditions for the BLUEs under the full restricted model to be the sums of BLUEs under the two small restricted models.

62H12 Estimation in multivariate analysis
62J05 Linear regression; mixed models
62J10 Analysis of variance and covariance (ANOVA)
Full Text: DOI
[1] Alalouf, IS; Styan, GPH, Characterizations of estimability in the general linear model, Ann Stat, 7, 194-200, (1979) · Zbl 0398.62053
[2] Bhimasankaram, P; Saharay, R, On a partitioned linear model and some associated reduced models, Linear Algebra Appl, 264, 329-339, (1997) · Zbl 0904.62078
[3] Chu, KL; Isotalo, J; Puntanen, S; Styan, GPH, On decomposing the Watson efficiency of ordinary least squares in a partitioned weakly singular linear model, Sankhyā Ser A, 66, 634-651, (2004) · Zbl 1193.62094
[4] Dong, B; Guo, W; Tian, Y, On relations between BLUEs under two transformed linear models, J Multivar Anal, 131, 279-292, (2014) · Zbl 1299.62055
[5] Drygas H (1970) The coordinate-free approach to Gauss-Markov estimation. Springer, Heidelberg · Zbl 0215.26504
[6] Groß, J; Puntanen, S, Estimations under a general partitioned linear model, Linear Algebra Appl, 321, 131-144, (2000) · Zbl 0966.62033
[7] Lu, C; Gan, S; Tian, Y, Some remarks on general linear model with new regressors, Stat Prob Lett, 97, 16-24, (2015) · Zbl 1312.62091
[8] Lu, C; Sun, Y; Tian, Y, On relations between weighted least-squares estimators of parametric functions under a general partitioned linear model and its small models, Metrika, 76, 707-722, (2013) · Zbl 1307.62180
[9] Marsaglia, G; Styan, GPH, Equalities and inequalities for ranks of matrices, Linear Multilinear Algebra, 2, 269-292, (1974) · Zbl 0297.15003
[10] Nurhonen, M; Puntanen, S, A property of partitioned generalized regression, Commun Stat Theory Methods, 21, 1579-1583, (1992) · Zbl 0800.62370
[11] Penrose, R, A generalized inverse for matrices, Proc Camb Phil Soc, 51, 406-413, (1955) · Zbl 0065.24603
[12] Rao, CR, Representations of best linear unbiased estimators in the Gauss-markoff model with a singular dispersion matrix, J Multivar Anal, 3, 276-292, (1973) · Zbl 0276.62068
[13] Tian, Y, More on maximal and minimal ranks of Schur complements with applications, Appl Math Comput, 152, 675-692, (2004) · Zbl 1077.15005
[14] Tian, Y, Some decompositions of OLSEs and BLUEs under a partitioned linear model, Int Stat Rev, 75, 224-248, (2007)
[15] Tian, Y, On an additive decomposition of the BLUE in a multiple partitioned linear model, J Multivar Anal, 100, 767-776, (2009) · Zbl 1155.62045
[16] Tian, Y, On equalities for BLUEs under misspecified Gauss-Markov models, Acta Math Sin Engl Ser, 25, 1907-1920, (2009) · Zbl 1180.62083
[17] Tian, Y; Beisiegel, M; Dagenais, E; Haines, C, On the natural restrictions in the singular Gauss-Markov model, Stat Pap, 49, 553-564, (2008) · Zbl 1148.62053
[18] Tian, Y; Takane, Y, On sum decompositions of weighted least-squares estimators for the partitioned linear model, Commun Stat Theor Methods, 37, 55-69, (2008) · Zbl 1139.62029
[19] Werner HJ, Yapar C (1995) More on partitioned possibly restricted linear regression. In: New trends in probability and statistics, Vol. 3: Multivariate statistics and matrices in statistics. In: Proceedings of the 5th conference, Tartu, Estonia, 23-28 May , 1994. Utrecht: VSP, pp 57-66. · Zbl 0844.62061
[20] Werner, HJ; Yapar, C, A BLUE decomposition in the general linear regression model, Linear Algebra Appl, 237, 395-404, (1996) · Zbl 0844.62061
[21] Zhang, B; Liu, B; Lu, C, A study of the equivalence of the BLUEs between a partitioned singular linear model and its reduced singular linear models, Acta Math Sin Engl Ser, 20, 557-568, (2004) · Zbl 1049.62083
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.