Principal components regression and r-k class predictions in linear mixed models

In this article, we propose the principal components regression and r-k class predictors, which combine the techniques of the ridge and principal components regressions in the linear mixed models. We demonstrate that the Henderson's predictors, the ridge predictors and the principal components regression predictors are special cases of the r-k class predictors. We also research assumption that the variance parameters are not known and get estimators of variance parameters. The necessary and sufficient conditions for the superiorities of the r-k class predictors over each of these three predictors are obtained by the criterion of mean square error matrix. Furthermore, we suggest tests to approve if these conditions are indeed satisfied. Finally, real data analysis and a simulation study are used to illustrate the findings.