CAREER: New Challenges in High-Dimensional and Nonparametric Statistics

复杂高维数据集的当代分析技术引起了统计学、数据科学和相关领域中许多基本概念的问题。在这个项目中，PI将解决在金融，工程和生命科学中的实际应用所激发的高维和非参数统计中具有挑战性的开放问题。该项目的重点是为复杂的数据集开发新的统计推断方法，提供高精度和明确的理论保证。这包括：（一）开发一个新的统计推断框架，这将大大扩展一些主要统计方法的适用范围;（二）在高维框架中研究恢复方法的性能;（三）研究高维模型的内在属性，以确保统计方法的良好性能。该项目的教育部分包括研究生和本科生的导师，面向STEM的高中生的统计和数据科学夏令营，以及初级研究人员的高维统计和学习理论研讨会/研究生院。该项目将特别注意支持代表性不足的少数群体的学生和研究人员，重点是两个主要研究主题。第一个主题是关于建立非渐近高阶扩展的概率分布之间的各种距离，特别侧重于在高维非渐近设置的问题和应用。PI将研究对于在高维中建立准确的近似边界至关重要的特性，例如正常近似和自举及其关系和最优性。该项目的另一个主要主题是开发一种基于非线性建模的统计推断新框架，并将其应用于非参数推断，功能估计和涉及重尾分布的模型推断。该方法结合了参数和非参数分量，可以避免严重的模型误设定，并建立良好的近似率。PI旨在对新的高阶近似界和非线性建模方法进行全面研究。该项目包括统计推断的新数学方法的开发，以及它们与其他数学领域（如高维概率、随机建模和不确定性量化）的联系的研究。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Contemporary techniques for analysis of complex high-dimensional data sets give rise to numerous questions about fundamental concepts in statistics, data science, and related fields. In this project, the PI will address challenging open questions in high-dimensional and nonparametric statistics motivated by practical applications in finance, engineering, and life sciences. The project is focused on development of new methods of statistical inference for complex data sets providing high accuracy and explicit theoretical guarantees. This includes (i) development of a novel framework for statistical inference that will considerably extend the range of applicability of some of the major statistical methods; (ii) studies of performance of resampling methods in a high-dimensional framework; and (iii) studies of intrinsic properties of high-dimensional models that ensure good performance of the statistical methods. The educational component of the project includes mentorship of graduate and undergraduate students, summer camps in statistics and data science for STEM-oriented high school students, and a workshop/graduate school on high-dimensional statistics and learning theory for junior researchers. Special attention will be given to supporting students and researchers from underrepresented minorities.The project is focused on two major research themes. The first theme is concerned with establishing non-asymptotic higher-order expansions for various distances between probability distributions, with a particular focus on problems and applications in a high-dimensional non-asymptotic setting. The PI will study characteristic properties that are crucial for establishing accurate approximation bounds in high dimensions, such as the normal approximation and bootstrapping and their relations and optimality properties. Another major theme of the project is development of a novel framework for statistical inference based on nonlinear modeling and its applications to nonparametric inference, functional estimation, and inference for models involving heavy-tailed distributions. The approach combines both parametric and nonparametric components, which can avoid severe model misspecification and establish good rates of approximation. The PI aims at conducting a comprehensive study of the new higher-order approximation bounds and the nonlinear modeling approach. The project includes development of new mathematical methods for statistical inference and studies of their connections with other areas of mathematics, such as high-dimensional probability, stochastic modeling, and uncertainty quantification.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.