Bayesian Composite Quantile Regression with Additive Regression Trees: A Robust Nonparametric Framework for Conditional Distribution Estimation

This paper introduces a comprehensive Bayesian nonparametric framework that unifies composite quantile regression with Bayesian additive regression trees (CQR-BART). The methodology addresses fundamental limitations in conventional quantile regression by simultaneously modeling multiple quantile levels while capturing complex nonlinear relationships through tree-based ensembles. We develop a hierarchical model specification using a location-scale mixture representation of the asymmetric Laplace distribution, which enables efficient Gibbs sampling via data augmentation techniques. Key theoretical contributions include establishing posterior consistency under mild regularity conditions and proving robustness properties through bounded influence functions. Extensive simulation studies across four challenging scenarios—homoscedastic nonlinear, heteroscedastic, heavy-tailed contaminated, and high-dimensional sparse settings—demonstrate superior performance compared to existing methods in terms of estimation accuracy, uncertainty quantification, and variable selection. Practical applications to economic forecasting (Growth-at-Risk analysis) and environmental data analysis (air pollution modeling) highlight the method's utility for robust conditional distribution estimation in real-world problems. The proposed approach is implemented in an accompanying \texttt{R} package \texttt{cqrbart}, ensuring reproducibility and accessibility for applied researchers.