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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.

##### MSC:
 62J12 Generalized linear models (logistic models)
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