Browsing by Subject "Quantile regression"

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.

Linear quantile regression is a powerful tool to investigate how predictors may affect a response heterogeneously across different quantile levels. Unfortunately, existing approaches find it extremely difficult to adjust for any dependency between observation units, largely because such methods are not based upon a fully generative model of the data. In this dissertation, we address this difficulty for analyzing spatial point-referenced data and hierarchical data. Several models are introduced by generalizing the joint quantile regression model of Yang and Tokdar (2017) and characterizing different dependency structures via a copula model on the underlying quantile levels of the observation units. A Bayesian semiparametric approach is introduced to perform inference of model parameters and carry out prediction. Multiple copula families are discussed for modeling response data with tail dependence and/or tail asymmetry. An effective model comparison criterion is provided for selecting between models with different combinations of sets of predictors, marginal base distributions and copula models.

Extensive simulation studies and real applications are presented to illustrate substantial gains of the proposed models in inference quality, prediction accuracy and uncertainty quantification over existing alternatives. Through case studies, we highlight that the proposed models admit great interpretability and are competent in offering insightful new discoveries of response-predictor relationship at non-central parts of the response distribution. The effectiveness of the proposed model comparison criteria is verified with both empirical and theoretical evidence.