Notes on Comparison of Covariance Matrices of BLUPs Under Linear Random-Effects Model with Its Two Subsample Models

A general linear random-effects model that includes both fixed and random effects, and its two subsample models are considered without making any restrictions on correlation of random effects and any full rank assumptions. Predictors of joint unknown parameter vectors under these three models have different algebraic expressions. Because of having different properties and performances under these models, it is one of the main focuses to make comparison of predictors. Covariance matrices of best linear unbiased predictors (BLUPs) of unknown parameters are used as a criterion to compare with other types predictors due to their definition of minimum covariance matrices structure. The comparison problem of covariance matrices of BLUPs under the models is considered in the study. We give a variety of equalities and inequalities in the comparison of covariance matrices of BLUPs of a general linear function of fixed effects and random effects under the models by using an approach consisting matrix rank and inertia formulas.