Unifying the prediction strategies of Theil-Goldberger and Kibria-Lukman within linear mixed models

Linear mixed models employ the best linear unbiased estimator and the best linear unbiased predictor to estimate the parameter vectors for fixed and random effects. However, due to the undesirable variance properties of the best linear unbiased estimator in the presence of multicollinearity, alternative estimators and predictors are preferred. The Theil-Goldberger’s and the Kibria-Lukman’s prediction approaches are commonly used for prediction under multicollinearity in linear mixed models. To address the issue of multicollinearity, this article introduces the mixed Kibria-Lukman estimator and predictor by combining these prediction approaches. To assess their effectiveness, the proposed mixed Kibria-Lukman estimator/predictor is compared with other estimators/predictors, including the best linear unbiased estimator/the best linear unbiased predictor and mixed estimators/predictors, using the matrix mean square error criterion. Furthermore, the performance of the newly defined prediction approach is demonstrated through the analysis of greenhouse gases data and a Monte-Carlo simulation study.