Foundations of High-Dimensional and Nonparametric Hypothesis Testing

Statistical inferential tools are the main export from the discipline of statistics to the empirical sciences, serving as the primary lens through which natural scientists interpret observations and quantify the uncertainty of their conclusions. However, in the analysis of modern large datasets the most common inferential tools available to us are fraught with pitfalls, often requiring various technical conditions to be checked before their valid application.

This in turn has led to misuse of the inferential tools and subsequent misinterpretation of results. This research project will aim to address this issue by developing and analyzing new user-friendly methodologies for statistical inference in complex settings. The methods we develop will be broadly applicable to a wide variety of challenging inferential problems in the physical and biological sciences, will eliminate the need to verify technical conditions, and will ultimately be robust in their application.

The principal and co-principal investigators will be involved in advising and mentoring graduate students, in curricular and course development, and in integrating the project with a research group on Statistical Methods in the Physical Sciences (STAMPS).

This project will advance our understanding of high-dimensional and non-parametric inference along three frontiers. Firstly, we aim to develop statistical inferential tools for irregular models, which are valid under weak conditions. Our particular focus will be on mixture models, and on methods which use sample-splitting to avoid strong regularity conditions.

Secondly, we will show that our methods achieve these strong guarantees at a surprisingly small statistical price. To rigorously quantify the statistical price paid for avoiding strong regularity conditions we will use minimax theory. However, standard minimax theory, in many cases, does not adequately capture the difficulty of statistical inference since the difficulty of inference can vary significantly across the parameter space.

A more refined theory -- called local minimax theory -- leads to a more accurate picture, and we will study our methods via this lens. Finally, we will address the problem of conditional independence (CI) testing. Despite its central role in regression diagnostics, and in the study of probabilistic graphical models, the task of CI testing and its intrinsic difficulty is poorly understood.

We will address two fundamental aspects of CI testing, by studying methods to appropriately calibrate CI tests, and by developing and analyzing powerful new CI tests.

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.