A Nonparametric Model-based Approach to Inference for Quantile Regression

In several regression applications, a different structural relationship might be anticipated for the higher or lower responses than the average responses. In such cases, quantile regression analysis can uncover important features that would likely be overlooked by traditional mean regression. We develop a Bayesian method for fully nonparametric model-based quantile regression. The approach involves flexible Dirichlet process mixture models for the joint distribution of the response and the covariates, with posterior inference for different quantile curves emerging from the conditional distribution of the response given the covariates. Inference is implemented using a combination of posterior simulation methods for Dirichlet process mixtures. Partially observed responses can also be handled within the proposed modeling framework leading to a novel nonparametric method for Tobit quantile regression. We use two data examples from the literature to illustrate the utility of the model, in particular, its capacity to uncover non-linearities in quantile regression curves as well as non-standard features in the response distribution.