Statistical inference of a partitioned linear random-effects model

Estimation and prediction of regression models with partition forms are widely applied techniques in statistical inference and data analysis. In this paper, we consider some fundamental inference problems regarding a linear random-effects model (LREM) and its reduced models without statistical distribution assumptions for error terms. We shall present a partition form of LREM and its correctly-reduced models, introduce the consistency concepts of the LREM and its reduced models, define the predictability/estimability of unknown parameters in the LREM and its reduced models, establish the matrix equations and analytical formulas associated with best linear unbiased predictors (BLUPs) and best linear unbiased estimators (BLUEs) of all unknown parameter vectors in the LREM and its reduced models, and present many fundamental decomposition equalities for the BLUPs/BLUEs of all unknown parameters in the LREM and its reduced models.