We introduce a semi-parametric Bayesian framework for a simultaneous analysis of linear quantile regression models. A simultaneous analysis is essential to attain the true potential of the quantile regression framework, but is computationally challenging due to the associated monotonicity constraint on the quantile curves. For a univariate covariate, we present a simpler equivalent characterization of the monotonicity constraint through an interpolation of two monotone curves. The resulting formulation leads to a tractable likelihood function and is embedded within a Bayesian framework where the two monotone curves are modeled via logistic transformations of a smooth Gaussian process. A multivariate extension is suggested by combining the full support univariate model with a linear projection of the predictors. The resulting single-index model remains easy to fit and provides substantial and measurable improvement over the first order linear heteroscedastic model. Two illustrative applications of the proposed method are provided.
"Simultaneous Linear Quantile Regression: A Semiparametric Bayesian Approach." Bayesian Anal. 7 (1) 51 - 72, March 2012. https://doi.org/10.1214/12-BA702