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On properties of BLUEs under general linear regression models. (English) Zbl 1428.62344
Summary: The best linear unbiased estimator (BLUE) of parametric functions of the regression coefficients under a general linear model \(\mathcal M=\{\mathbf {y,X\beta} ,\sigma ^{2}\mathbf {\Sigma }\}\) can be written as \(\mathbf{Gy}\), where \(\mathbf{G}\) is the solution of a consistent linear matrix equation composed by the given matrices in the model and their generalized inverses. In the past several years, a useful tool – the matrix rank method was utilized to simplify various complicated operations of matrices and their generalized inverses. In this paper, we use this algebraic method to give a comprehensive investigation to various algebraic and statistical properties of the projection matrix \(\mathbf{G}\) in the BLUE of parametric functions under \(\mathcal M\). These properties include the uniqueness of \(\mathbf{G}\), the maximal and minimal possible ranks of \(\mathbf{G}\) and \(\mathrm{Cov}(\mathbf{Gy})\), as well as identifying conditions for various equalities for \(\mathbf{G}\). In addition, necessary and sufficient conditions were established for equalities of projection matrices in the BLUEs of parametric functions under the original model and its transformed models.

62J12 Generalized linear models (logistic models)
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