Pyramid quantile regression

Quantile regression models provide a wide picture of the conditional distributions of the response variable by capturing the effect of the covariates at different quantile levels. In most applications, the parametric form of those conditional distributions is unknown and varies across the covariate space, so fitting the given quantile levels simultaneously without relying on parametric assumptions is crucial. In this work we propose a Bayesian model for simultaneous linear quantile regression. More specifically, we propose to model the conditional distributions by using random probability measures known as quantile pyramids. Unlike many existing approaches, our framework allows us to specify meaningful priors on the conditional distributions, whilst retaining the flexibility afforded by the nonparametric error distribution formulation. Simulation studies demonstrate the flexibility of the proposed approach in estimating diverse scenarios, generally outperforming other competitive methods. The method is particularly promising for modelling the extremal quantiles. Applications to linear splines and extreme value analysis are also explored through real data examples.