A study of comparison problems on linear experiments with stochastic regression coefficients

Prediction and estimation of unknown parameters and comparison issues between two linear models have always been research topics of great interest to statisticians. Since the comparison problem of linear models was proposed in the 1950s, many statisticians have conducted relevant research. The two models are compared based on certain optimal standards. The comparison of two random effects models where covariance matrices is known and symmetric positive semi-definite is considered. The paper characterizes the following relations: one random effect model at least as good as the other, better strictly than the other, and equivalent to the other in comparing with the covariance matrix of the best linear unbiased predictor of the linear predictable function of the regression coefficient, and gives out the necessary and sufficient conditions respectively by using the effective matrix analysis tools. The theoretical basis can be determined by them for model selection during statistical modeling. Finally, the comparison problem between two general linear models is considered; when covariance matrices are non singular, the relevant topic is also studied. As an application, a case on the comparison of two linear models with linear parameter restrictions is given to illustrate the main results.