On additive decompositions of estimators under a multivariate general linear model and its two submodels

Parameters from linear regression models are often estimated by the ordinary least squares estimator (OLSE) or by the best linear unbiased estimator (BLUE). These estimators can be written in analytical form, so that it is not difficult to describe their performances under various model assumptions. In this paper, we study the problem of additive decompositions of OLSEs and BLUEs of parameter spaces in a full multivariate general linear model (MGLM) and in two specific submodels. We establish necessary and sufficient conditions for the validity of various identities involving the OLSEs and BLUEs of whole and partial mean parameter matrices under the MGLM and two smaller MGLMs.