Some new developments for quantile regression

Abstract

Quantile regression (QR) (Koenker and Bassett, 1978), as a comprehensive extension to standard mean regression, has been steadily promoted from both theoretical and applied aspects. Bayesian quantile regression (BQR), which deals with unknown parameter estimation and model uncertainty, is a newly proposed tool of QR. This thesis aims to make some novel contributions to the following three issues related to QR. First, whereas QR for continuous responses has received much attention in literatures, QR for discrete responses has received far less attention. Second, conventional QR methods often show that QR curves crossing lead to invalid distributions for the response. In particular, given a set of covariates, it may turn out, for example, that the predicted 95th percentile of the response is smaller than the 90th percentile for some values of the covariates. Third, mean-based clustering methods are widely developed, but need improvements to deal with clustering extreme-type, heavy tailed-type or outliers problems. This thesis focuses on methods developed over these three challenges: modelling quantile regression with discrete responses, ensuring non-crossing quantile curves for any given sample and modelling tails for collinear data with outliers. The main contributions are listed as below: * The first challenge is studied in Chapter 2, in which a general method for Bayesian inference of regression models beyond the mean with discrete responses is developed. In particular, this method is developed for both Bayesian quantile regression and Bayesian expectile regression. This method provides a direct Bayesian approach to these regression models with a simple and intuitive interpretation of the regression results. The posterior distribution under this approach is shown to not only be coherent to the response variable, irrespective of its true distribution, but also proper in relation to improper priors for unknown model parameters. * Chapter 3 investigates a new kernel-weighted likelihood smoothing quantile regression method. The likelihood is based on a normal scale-mixture representation of an asymmetric Laplace distribution (ALD). This approach benefits of the same good design adaptation just as the local quantile regression (Spokoiny et al., 2014) does and ensures non-crossing quantile curves for any given sample. * In Chapter 4, we introduce an asymmetric Laplace distribution to model the response variable using profile regression, a Bayesian non-parametric model for clustering responses and covariates simultaneously. This development allows us to model more accurately for clusters which are asymmetric and predict more accurately for extreme values of the response variable and/or outliers. In addition to the three major aforementioned challenges, this thesis also addresses other important issues such as smoothing extreme quantile curves and avoiding insensitive to heteroscedastic errors as well as outliers in the response variable. The performances of all the three developments are evaluated via both simulation studies and real data analysis.