Novel Inference Procedures for Non-Standard High-Dimensional Regression Models

假设检验的统计理论在几乎所有科学研究中都起着基础性的作用。在大数据时代，高维数据在自然科学、社会科学、医学、公共卫生等多个科学领域无处不在。因此，现代应用往往涉及到高维下的假设检验，这就需要新的统计推断理论。由于回归是应用最广泛的统计分析工具，最近的一些工作集中在高维最小二乘回归的假设检验上。然而，众所周知，标准的最小二乘回归模型在实际应用中存在严重的局限性。本研究旨在为更具弹性的高维回归模型发展新的统计推断理论。本文主要研究超高维下几种重要的非标准回归模型的推理理论的发展。具体地说，PI将开发三种模型下的线性假设测试：高维预期回归、高维异方差回归和稳健高维回归。测试统计量的渐近分布将被严格地建立。理论研究将填补高维统计文献中的重要空白。将开发一种统一的高效算法来解决计算挑战。这项研究将为研究预期函数、检验高维数据的异质性和进行稳健推理提供有原则的工具。将为研究生提供研究培训机会。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Statistical theory of hypothesis testing plays a fundamental role in virtually all scientific studies. In the era of big data, high-dimensional data are ubiquitous in many scientific fields such as natural sciences, social sciences, medicine, and public health. Therefore, modern applications often involve hypothesis testing under high dimensions, which calls for new statistical inference theory. As regression is the most popular statistical analysis tool in applications, some recent work has been focused on hypothesis testing in high dimensional least squares regression. However, it is well-known that the standard least squares regression model has severe limitations in real applications. This research aims to develop new statistical inference theory for more flexible high dimensional regression models. This research focuses on the development of inference theory for several important non-standard regression models under ultra-high dimensions. Specifically, the PI will develop tests for testing linear hypotheses under three models: high dimensional expectile regression, high dimensional heteroscedastic regression, and robust high dimensional regression. Asymptotic distributions of the test statistics will be established rigorously. The theoretical study will fill important gaps in the high-dimensional statistics literature. A unified efficient algorithm will be developed to tackle the computational challenges. The research will provide principled tools for studying expectile functions, for examining the heterogeneity in high dimensional data and for performing robust inference. Research training opportunities for graduate students will be provided.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.