<p>In statistical theory and applications of parametric regression models, the unknown parameters in a given parametric model may satisfy certain pre-assumed restrictions, and thereby it is a common practice to add the restrictions in the estimation procedures of the unknown parameters. In this article, we assume that the unknown parameters in a general linear model satisfy two competing linear restriction equations, and reconsider a comparison problem in the analysis of covariance matrices of the best linear unbiased predictors (BLUPs) and the best linear unbiased estimators (BLUEs) under the two competing constrained linear models using some new mathematical analysis tools. We shall give analytic expressions of BLUPs and BLUEs of a joint vector of all unknown parameters in the two competing constrained linear models, present some of their algebraic and statistical properties, and establish a diverse of analytical formulas for calculating the ranks and inertias of the differences of the covariance matrices of the two kinds of predictors and estimators of all unknown parametric functions under two constrained linear models. We also derive necessary and sufficient conditions for a sequence of equalities and inequalities for the covariance matrices of BLUPs and BLUEs to hold under the two competing constrained linear models. Finally, an example based on an agricultural block-treatment design is used to illustrate the theoretical results.</p>