A study of the equivalence of inference results in the contexts of true and misspecified multivariate general linear models

In practical applications of regression models, we may meet with the situation where a true model is misspecified in some other forms due to certain unforeseeable reasons, so that estimation and statistical inference results obtained under the true and misspecified regression models are not necessarily the same, and therefore, it is necessary to compare these results and to establish certain links between them for the purpose of reasonably explaining and utilizing the misspecified regression models. In this paper, we propose and investigate some comparison and equivalence analysis problems on estimations and predictions under true and misspecified multivariate general linear models. We first give the derivations and presentations of the best linear unbiased predictors (BLUPs) and the best linear unbiased estimators (BLUEs) of unknown parametric matrices under a true multivariate general linear model and its misspecified form. We then derive a variety of necessary and sufficient conditions for the BLUPs/BLUEs under the two competing models to be equal using a series of highly-selective formulas and facts associated with ranks, ranges and generalized inverses of matrices, as well as block matrix operations.