In this study, we propose a Bayesian Empirical Likelihood (BEL) method for linear mixed-effects models by integrating Huber-type influence functions and random effects annihilation matrices. At the group-level, we constructed several interpretable moment conditions, including robust residual balance moments and slope-related moments obtained after eliminating random intercepts through the annihilation matrix. We employed a block-diagonal weighting matrix to harmonize scaling across moment blocks. The BEL posterior distribution was derived by combining empirical likelihood with the prior distribution, and a Metropolis-Hastings algorithm for practical computation was designed. In simulation studies under adverse conditions, the finite-sample behavior of the proposed BEL method was compared with that of standard empirical likelihood (EL), restricted maximum likelihood (REML), maximum likelihood (ML), fully parametric Bayesian estimation (Bayes), and generalized moment estimation (GMM). The results showed that under correctly specified Gaussian LMMs, BEL attains efficiency comparable to parametric methods; whereas under adverse conditions, BEL generally achieves smaller mean squared errors than other methods, and its robustness is comparable to GMM. The application of real panel data further demonstrated that BEL can mitigate the impact of observations in extreme regions while preserving the major trend relationships, thus obtaining a more stable and reliable fixed effects estimate in real-world situations where model misspecification and atypical observations are difficult to avoid.
Citation: Youxi Luo, Shiqi Zhou, Chaozhu Hu, Hanfang Li. Robust Bayesian empirical likelihood estimation for linear mixed-effects models[J]. AIMS Mathematics, 2026, 11(3): 6955-6988. doi: 10.3934/math.2026286
In this study, we propose a Bayesian Empirical Likelihood (BEL) method for linear mixed-effects models by integrating Huber-type influence functions and random effects annihilation matrices. At the group-level, we constructed several interpretable moment conditions, including robust residual balance moments and slope-related moments obtained after eliminating random intercepts through the annihilation matrix. We employed a block-diagonal weighting matrix to harmonize scaling across moment blocks. The BEL posterior distribution was derived by combining empirical likelihood with the prior distribution, and a Metropolis-Hastings algorithm for practical computation was designed. In simulation studies under adverse conditions, the finite-sample behavior of the proposed BEL method was compared with that of standard empirical likelihood (EL), restricted maximum likelihood (REML), maximum likelihood (ML), fully parametric Bayesian estimation (Bayes), and generalized moment estimation (GMM). The results showed that under correctly specified Gaussian LMMs, BEL attains efficiency comparable to parametric methods; whereas under adverse conditions, BEL generally achieves smaller mean squared errors than other methods, and its robustness is comparable to GMM. The application of real panel data further demonstrated that BEL can mitigate the impact of observations in extreme regions while preserving the major trend relationships, thus obtaining a more stable and reliable fixed effects estimate in real-world situations where model misspecification and atypical observations are difficult to avoid.
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Youxi Luo, Shiqi Zhou, Chaozhu Hu, Hanfang Li. Robust Bayesian empirical likelihood estimation for linear mixed-effects models[J]. AIMS Mathematics, 2026, 11(3): 6955-6988. doi: 10.3934/math.2026286
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