Model selection in quantile regression models

Lasso methods are regularization and shrinkage methods widely used for subset selection and estimation in regression problems. From a Bayesian perspective, the Lasso-type estimate can be viewed as a Bayesian posterior mode when specifying independent Laplace prior distributions for the coefficients of independent variables (Park and Casella, 2008). A scale mixture of normal priors can also provide an adaptive regularization method and represents an alternative model to the Bayesian Lasso-type model. In this paper, we assign a normal prior with mean zero and unknown variance for each quantile coefficient of independent variable. Then, a simple MCMC-based computation technique is developed for quantile regression models, including continuous, binary and left-censored outcomes. Based on the proposed prior, we propose a criterion for model selection in quantile regression models. The proposed criterion can be applied to classical least squares, classical quantile regression (QReg), classical Tobit QReg and many others. For example, the proposed criterion can be applied to rq(), lm() and crq() which is available in an R package called Brq. Through simulation studies and analysis of a prostate cancer dataset, we assess the performance of the proposed methods. The simulation studies and the prostate cancer dataset analysis confirm that our methods perform well, compared to other approaches.