On properties of BLUEs under general linear regression models.

*(English)*Zbl 1428.62344Summary: 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) |