New Methods for Bayesian Quantile Regression Modeling

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This research project will develop flexible models and corresponding methods for inference and variable selection for a number of quantile regression problems. Regression modeling to explore the effect of a number of predictor variables on a response variable remains a key activity in statistical research. By targeting different percentiles of the response variable, quantile regression provides a practically important alternative to traditional mean regression for settings where the predictors may affect the tails of the response distribution in a different fashion than its center. This project will advance statistical methodology within the composite quantile modeling framework. Quantile regression is a burgeoning area of statistical research with great promise for substantive applications. The project will investigate the application of the methods to several problems within the economic and social sciences. The methods will be used, for example, to study the relative importance of different types of predictors on different levels of mathematical achievement for high school students. To facilitate use of the methods by practitioners and researchers from other fields, publicly available software will be developed for implementing several of the models. The project will create educational opportunities for graduate students and seek to increase the participation of women and underrepresented groups in the research.This research project will develop a general model-based framework for composite quantile regression to allow estimation and selection of predictor effects to be informed by multiple percentiles of the response distribution. The composite quantile regression modeling framework involves structured mixtures of distributions that will facilitate different inferential objectives in the integration of effects from several predictor variables through multiple response quantiles. Composite quantile regression is a recent development that has not yet been studied under a probabilistic modeling framework for the response distribution. The research project has a substantial analytic component with regard to the theoretical study of properties for the various quantile regression models as well as a challenging computational component with regard to achieving computationally tractable model fitting. The practical utility of the new methods will be investigated through several simulation studies in the context of variable selection to compare with well-established regularized regression methods. For all the models, the project will involve the study of relevant theoretical properties and design of computational techniques for inference, model checking, and prediction. The project also will explore hierarchical model extensions for practically relevant settings with multilevel data structures.