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This paper presents a Bayesian analysis of partially linear additive models for quantile regression. We develop a semiparametric Bayesian approach to quantile regression models using a spectral representation of the nonparametric regression functions and the Dirichlet process (DP) mixture for error distribution. We also consider Bayesian variable selection procedures for both parametric and nonparametric components in a partially linear additive model structure based on the Bayesian shrinkage priors via a stochastic search algorithm. Based on the proposed Bayesian semiparametric additive quantile regression model referred to as BSAQ, the Bayesian inference is considered for estimation and model selection. For the posterior computation, we design a simple and efficient Gibbs sampler based on a location-scale mixture of exponential and normal distributions for an asymmetric Laplace distribution, which facilitates the commonly used collapsed Gibbs sampling algorithms for the DP mixture models. Additionally, we discuss the asymptotic property of the sempiparametric quantile regression model in terms of consistency of posterior distribution. Simulation studies and real data application examples illustrate the proposed method and compare it with Bayesian quantile regression methods in the literature. [ABSTRACT FROM AUTHOR]

Mixture models provide a method of modelling a complex probability distribution in terms of simpler structures. In particular, the method of mixture of regressions has received considerable attention due to its modelling flexibility and availability of convenient computational algorithms. This paper aims to contribute to theoretical justification for the mixtures of regression model from the Bayesian perspective. In particular, we establish consistency of posterior distribution and determine how fast posterior distribution converges to the true value of the parameter in the context of mixture of binary, Poisson, and Gaussian regressions. [ABSTRACT FROM AUTHOR]