New Methods for Bayesian Quantile Regression Modeling

该研究项目将为许多分位数回归问题开发灵活的模型和相应的推理和变量选择方法。用于探索多个预测变量对响应变量的影响的回归模型仍然是统计研究中的关键活动。通过针对响应变量的不同百分位数，分位数回归为预测变量可能以与响应分布中心不同的方式影响响应分布尾部的设置提供了传统均值回归的实际重要替代方案。该项目将在复合分位数建模框架内推进统计方法。分位数回归是统计研究的一个新兴领域，具有巨大的实质性应用前景。该项目将研究这些方法在经济和社会科学中几个问题的应用。例如，这些方法将用于研究不同类型的预测因素对高中生不同数学成绩水平的相对重要性。为了方便其他领域的从业者和研究人员使用这些方法，将开发公开可用的软件来实施多个模型。该项目将为研究生创造教育机会，并寻求增加女性和代表性不足群体在研究中的参与。该研究项目将为复合分位数回归开发一个基于模型的通用框架，以便根据响应分布的多个百分位数来估计和选择预测效应。复合分位数回归建模框架涉及分布的结构化混合，这将有助于在通过多个响应分位数整合多个预测变量的影响时实现不同的推理目标。复合分位数回归是最近的发展，尚未在响应分布的概率建模框架下进行研究。该研究项目具有关于各种分位数回归模型属性的理论研究的实质性分析部分，以及关于实现计算上易于处理的模型拟合的具有挑战性的计算部分。将通过变量选择背景下的多项模拟研究来研究新方法的实际效用，以便与成熟的正则化回归方法进行比较。对于所有模型，该项目将涉及相关理论特性的研究以及用于推理、模型检查和预测的计算技术的设计。该项目还将探索分层模型扩展，以实现具有多级数据结构的实际相关设置。

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.