  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.
