CAREER: New Challenges in High-Dimensional and Nonparametric Statistics

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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.