Some Explorations of Bayesian Joint Quantile Regression

Abstract

Although quantile regression provides a comprehensive and robust replacement for the traditional mean regression, a complete estimation technique is in blank for a long time. Original separate estimation could cause severe problems, which obstructs its popularization in methodology and application. A novel complete Bayesian joint estimation of quantile regression is proposed and serves as a thorough solution to this historical challenge. In this thesis, we first introduce this modeling technique and propose some preliminary but important theoretical development on the posterior convergence rate of this novel joint estimation, which offers significant guidance to the ultimate results. We provide the posterior convergence rate for the density estimation model induced by this joint quantile regression model. Furthermore, the prior concentration condition of the truncated version of this joint quantile regression model is proved and the entropy condition of the truncated model with any sphere predictor plane centered at 0 is verified. An application on high school math achievement is also introduced, which reveals some deep association between math achievement and socio-economic status. Some further developments about the estimation technique, convergence rate and application are discussed. Furthermore, some suggestions on school choices for minority students are mentioned according to the application.