Quantile pyramids for 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 talk, I introduce a Bayesian model for simultaneous quantile regression. More specifically, I propose to model the conditional distributions using random probability measures known as quantile pyramids. Unlike many existing approaches, this 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 both linear and splines quantile regression models. The method is particularly promising for modelling the extremal quantiles, where it significantly outperforms other competitive approaches.