On comparison of dispersion matrices of estimators under a constrained linear model

Abstract

We introduce some new mathematical tools in the analysis of dispersion matrices of the two well-known OLSEs and BLUEs under general linear models with parameter restrictions. We first establish some formulas for calculating the ranks and inertias of the differences of OLSEs’ and BLUEs’ dispersion matrices of parametric functions under the general linear model ${\mathscr {M}}= \{\mathbf{y}, \ \mathbf{X }\pmb {\beta }, \ \pmb {\Sigma }\}$ and the constrained model ${\mathscr {M}}_r = \{\mathbf{y}, \, \mathbf{X }\pmb {\beta }\, | \, \mathbf{A }\pmb {\beta }= \mathbf{b}, \ \pmb {\Sigma }\}$, where $\mathbf{A }\pmb {\beta }= \mathbf{b}$ is a consistent linear matrix equation for the unknown parameter vector $\pmb {\beta }$ to satisfy. As applications, we derive necessary and sufficient conditions for many equalities and inequalities of OLSEs’ and BLUEs’ dispersion matrices to hold under ${\mathscr {M}}$ and ${\mathscr {M}}_r$.