Inference for High Dimensional Quantile Regression

在多种应用的推动下，分位数回归正在成为一个重要而活跃的研究领域。与传统的最小二乘回归方法相比，分位数回归方法对异常值具有较强的稳健性，能够捕捉异质性。近年来，关于分位数回归估计和变量选择的研究在大数据应用方面取得了显著的成果。然而，关于高维环境下分位数回归的假设检验和可信区间构建等推理方法的研究很少。这个项目寻求开发新的统计理论、方法和算法来解决高维分位数回归中的推断问题。这项研究是由糖尿病干预的大型临床试验推动的，所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高维协变量在响应分布的分位数上的总体重要性。提出了两种检验方法，一种是基于边际分位数回归的最大值统计量，另一种是记分型统计量。将研究固定维度和发散维度的理论和方法。其次，PI将专注于高维分位数回归，而不是传统的关于系数的最小信号强度条件，它将严格研究惩罚估计的渐近理论，并基于渐近理论和Bootstrap过程发展有效的选择后推断方法。国际和平研究所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，并通过合作和知识共享，接触K-12学生和发展中国家，从而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.