Nonparametric Bayesian Quantile Regression via Dirichlet Process Mixture Models

Abstract

We propose new nonparametric Bayesian approaches to quantile regression using Dirichlet process mixture (DPM) models. All the existing quantile regression methods based on DPMs require the kernel density to satisfy the quantile constraint, hence the kernel densities are themselves usually in the form of mixtures. One innovation of our approaches is that we impose no constraint on the kernel, thus a wide range of densities can be chosen as the kernels of the DPM model. The quantile constraint is satisfied by a post-processing of the DPM by a suitable location shift. As a result, our proposed models use simpler kernels and yet possess great flexibility by mixing over both the location parameter and the scale parameter. The posterior consistency of our proposed model is studied carefully. And Markov chain Monte Carlo algorithms are provided for posterior inference. The performance of our approaches is evaluated using simulated data and real data. Moreover, we are able to incorporate random effects into our models such that our approaches can be extended to handle longitudinal data.