A further prediction method in linear mixed models: Liu prediction

Abstract

We propose the Liu estimator and the Liu predictor via the penalized log-likelihood approach in linear mixed models when multicollinearity is present. The necessary and sufficient conditions for the superiority of the Liu predictor over the best linear unbiased predictor and the ridge predictor of linear combinations of fixed and random effects in the sense of matrix and scalar mean square errors are examined. Furthermore, the selection of the Liu biasing parameter is given and the findings are illustrated with both a real data set and a simulation study. The study show that the Liu estimator and predictor outperform the ridge estimator and predictor and the blue and blup in the sense of mean square error for large degree of correlation and the degree of supremacy of the Liu estimator and predictor over the ridge estimator and predictor and the blue and blup increase as the Liu biasing parameter decreases.