Shrinkage estimation of fixed and random effects in linear quantile mixed models

Area of Science:

• Statistics
• Econometrics
• Biostatistics

Background:

• Linear mixed models are widely used for analyzing correlated data.
• Quantile regression is essential for understanding conditional distribution tails.
• Integrating Bayesian inference with mixed models and quantile regression presents computational challenges.

Purpose of the Study:

• To develop a Bayesian framework for linear mixed quantile regression.
• To introduce novel Bayesian shrinkage priors for fixed and random effects.
• To extend the methodology to Bayesian mixed expectile models.

Main Methods:

• Utilized a modified Cholesky decomposition for the random effects covariance matrix.
• Employed an asymmetric Laplace distribution for error modeling.
• Developed a partially collapsed Gibbs sampling algorithm for efficient posterior inference and a Metropolis-Hastings acceptance-rejection (MHAR) algorithm for expectile models.

Main Results:

• The proposed Bayesian shrinkage approaches effectively regularize model parameters.
• The partially collapsed Gibbs sampler significantly improves Markov chain mixing.
• The extended expectile model framework and MHAR algorithm provide robust estimation.

Conclusions:

• The novel Bayesian approach offers a flexible and computationally efficient tool for mixed quantile and expectile regression.
• The method demonstrates superior performance compared to existing approaches in both simulated and real data.
• This work advances the application of Bayesian statistics in complex regression modeling.