Analysis of best linear unbiased predictions in the contexts of a linear mixed model and its six correctly-reduced models

A parametric regression model can be decomposed into certain partitioned forms, consequently, we are able to construct some small or reduced parametric models that have the same partial unknown parameter vectors with the partitioned model. In this case, predictions/estimations of the same partial unknown parameters in the contexts of the original and reduced models may have different expressions and performances, and thus it would be of interest to consider the relationship for these inference results. This is known to be a tedious task because expressions of prediction/estimation results involve a diverse of complicated matrix calculations. This paper reconsiders this subject in the contexts of a partitioned linear mixed model and its six correctly-reduced linear mixed models (CRLMMs). We shall develop a general theory of establishing connections between the best linear unbiased predictors of the same partial unknown parameters under a partitioned linear mixed model and its correctly-reduced linear mixed models using a series of precise matrix analysis tools. The results obtained in the paper are suitable for other special partitioned linear mixed models, and thus this methodological or theoretical innovation work can serve as general reference on relations research between the predictors under more complicated linear models.