Inference for High Dimensional Quantile Regression

分位数回归在多种应用的推动下，成为一个重要而活跃的研究领域。与传统的最小二乘回归相比，分位数回归方法对异常值具有鲁棒性，可以捕获异质性。近年来，在大数据应用中，分位数回归的估计和变量选择等方面取得了重要成果。然而，对于高维环境下分位数回归的假设检验和置信区间构建等推理方法的研究却很少。本项目旨在发展新的统计理论、方法和算法，以解决高维分位数回归中的推理问题。这项研究的动机是糖尿病干预的大型临床试验，开发的方法也可以应用于全基因组关联研究、神经科学和环境研究的数据。研究将集中在两个主要方向。首先，将开发新的测试程序来评估响应分布的分位数上高维协变量的总体意义。提出了两种类型的检验，包括基于边际分位数回归的最大型统计量和分数型统计量。将研究固定维数和发散维数的理论和方法。其次，重点是在系数上没有传统的最小信号强度条件的高维分位数回归，PI将严格研究惩罚估计量的渐近理论，并基于渐近理论和自举过程开发有效的后选择推理方法。该项目将通过开发高级专题课程、吸引研究生和本科生（特别是来自代表性不足群体的学生）参与该项目、通过合作和知识共享向K-12学生和发展中国家提供服务等方式，将研究与教育结合起来。

Quantile regression is emerging as an important and active research area driven by diverse applications. Compared with the conventional least squares regression, quantile regression methods are robust against outliers and can capture heterogeneity. In recent years, significant results related to estimation and variable selection for quantile regression have been obtained for big data applications. However, there exists little work on inferential methods including hypothesis testing and confidence interval construction for quantile regression in the high-dimensional setting. This project seeks to develop new statistical theory, methodology and algorithms to address inference problems in high-dimensional quantile regression. The research is motivated by a large clinical trial of diabetes intervention, and the developed methods can also be applied to data from genome-wide association studies, neuroscience, and environmental studies. The research will focus on two main directions. First, new testing procedures will be developed to assess the overall significance of high-dimensional covariates on quantiles of the response distribution. Two types of tests are proposed, including a maximum-type statistic based on marginal quantile regression, and a score-type statistic. Theory and methods for both fixed and diverging dimensions will be studied. Second, focusing on high-dimensional quantile regression without the conventional minimum signal strength condition on the coefficients, the PI will rigorously study the asymptotic theory of penalized estimators and develop valid post-selection inference methods based on both asymptotic theory and bootstrap procedures. The PI will integrate research and education by developing advanced topics courses, engaging graduate and undergraduate students, especially those from under-represented groups, in the project, and reaching out to K-12 students and developing countries through collaboration and knowledge sharing.