Bayesian Quantile Regression Based on the Empirical Likelihood with Spike and Slab Priors

Abstract

In this paper, we consider nonparametric Bayesian variable selection in quantile regression. The Bayesian model is based on the empirical likelihood, and the prior is chosen as the “spike-and-slab” prior–a mixture of a point mass at zero and a normal distribution. We show that the posterior distribution of the zero coefficients converges to a point mass at zero and that of the nonzero coefficients converges to a normal distribution. To further address the problem of low statistical efficiency in extreme quantile regression, we extend the Bayesian model such that it can integrate information at multiple quantiles to provide more accurate inference of extreme quantiles for homogenous error models. Simulation studies demonstrate that the proposed methods outperform or perform equally well compared with existing methods. We apply this Bayesian method to study the role of microRNAs on regulating gene expression and find that the regulation of microRNA may have a positive effect on the gene expression variation.