, Volume 81, Issue 5, pp 569–586 | Cite as

Shrinkage estimation in linear mixed models for longitudinal data

  • Shakhawat Hossain
  • Trevor Thomson
  • Ejaz Ahmed


This paper is concerned with the selection and estimation of fixed effects in linear mixed models while the random effects are treated as nuisance parameters. We propose the non-penalty James–Stein shrinkage and pretest estimation methods based on linear mixed models for longitudinal data when some of the fixed effect parameters are under a linear restriction. We establish the asymptotic distributional biases and risks of the proposed estimators, and investigate their relative performance with respect to the unrestricted maximum likelihood estimator (UE). Furthermore, we investigate the penalty (LASSO and adaptive LASSO) estimation methods and compare their relative performance with the non-penalty pretest and shrinkage estimators. A simulation study for various combinations of the inactive covariates shows that the shrinkage estimators perform better than the penalty estimators in certain parts of the parameter space. This particularly happens when there are many inactive covariates in the model. It also shows that the pretest, shrinkage, and penalty estimators all outperform the UE. We further illustrate the proposed procedures via a real data example.


Asymptotic distributional bias and risk Linear mixed model Likelihood ratio test LASSO Monte Carlo simulation Shrinkage and pretest estimators 



We would like to thank the referees, editor and associate editor for their valuable suggestions in the revision of this paper. The research of Shakhawat Hossain and Ejaz Ahmed was supported by the Natural Sciences and the Engineering Research Council of Canada.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of WinnipegWinnipegCanada
  2. 2.Department of Mathematics and StatisticsBrock UniversitySt. CatharinesCanada

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