Pyramid Quantile Regression

Abstract

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. These models have been applied to several areas, including the environmental sciences, medicine, engineering, and economics. 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. However, most quantile regression models only provide individual quantile fitting, so final estimates are generally dissociated from one another and may not respect the monotonicity constraint of the quantile function.

In this thesis, we first propose a two-stage approach to overcome the crossing problem in quantile regression, where separately fitted curves for several quantiles may cross. The postprocessing step is carried out via Gaussian process regression over the initial quantile estimates. We demonstrate that by borrowing strength from nearby quantiles, we can monotonize the quantile function and add smoothness to the quantile process, while maintaining posterior consistency properties of the initial estimates.

Although postprocessing is an interesting tool for correcting the crossing, it does not deal with other issues related to individual quantile fitting, like excessively wiggly fitted curves and poor accuracy, specially for tail quantile estimates. In order to tackle that, we also propose a Bayesian model for simultaneous quantile regression using random probability measures known as quantile pyramids. Unlike many existing approaches, our framework allows us to specify meaningful priors directly on the conditional distributions, whilst retaining the flexibility afforded by the nonparametric quantile distribution formulation. Posterior consistency results are presented, and small sample properties are investigated through simulation studies.

Finally, we also provide a fully nonparametric modelling framework for simultaneous estimation of quantile curves. Quantile pyramids are again employed as a prior for the conditional distributions, and an attractive formulation is considered to automatically impose monotonicity constraints. The nonparametric curves are modelled using cubic B-splines, and a penalisation approach is considered to tackle the challenges of unknown number and location of knots.