Bayesian Regularized Quantile Regression Using Adaptive Lasso and Its Applications

Bayesian regularized procedures for parameter estimation and model selection in quantile regression models have received much attention in the literature. This work continues the line of research by proposing a novel Bayesian adaptive Lasso quantile regression procedure for the analysis of data with non-ignorable missing responses. Specifically, we present a Bayesian hierarchical setting that allows for specifications of different penalization parameters for the regression coefficients in quantile regression models with asymmetric Laplace density and a specification of the logistic regression model for dealing with the non-ignorable missing mechanism. Thanks to the normal-exponential mixture representation of the asymmetric Laplace distribution and Student-t approximation of the logistic regression model, we propose an efficient Markov Chain Monte Carlo sampling algorithm for making posterior inferences. We then generalize the proposed Bayesianprocedure to simultaneously perform variable selection and coefficient estimation for analyzing longitudinal data with missing responses. This is done by introducing a semiparametric mixed-effects double quantile regression model to jointly model the mean and variance of the mixed-effects as functions of the predictors. Finally, we develop a Bayesian adaptive Lasso procedure to tackle the change-point detection problem, while accounting for issues, such as serial correlation, non-normality of data, and the presence of linear trends in data series. To be more specific, we deal with these issues by treating multiple change-point detection as a model selection problem in a linear quantile regression model with autoregressive errors. We then propose an efficient Bayesian sampling procedure with good convergence and mixing properties for exploring the full posterior distributions of the unknown parameters.