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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.

MSC:
62H12 Estimation in multivariate analysis
62J05 Linear regression; mixed models
62J10 Analysis of variance and covariance (ANOVA)
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