Collaborative Research: Synergies between Steins Identities and Reproducing Kernels: Modern Tools for Nonparametric Statistics

The project aims to conduct comprehensive statistical and computational analyses, with the overarching objective of advancing innovative nonparametric data analysis techniques. The methodologies and theories developed are anticipated to push the boundaries of modern nonparametric statistical inference and find applicability in other statistical domains such as nonparametric latent variable models, time series analysis, and sequential nonparametric multiple testing. This project will enhance the interconnections among statistics, machine learning, and computation and provide training opportunities for postdoctoral fellows, graduate students, and undergraduates. More specifically, the project covers key problems in nonparametric hypothesis testing, intending to establish a robust framework for goodness-of-fit testing for distributions on non-Euclidean domains with unknown normalization constants. The research also delves into nonparametric variational inference, aiming to create a particle-based algorithmic framework with discrete-time guarantees. Furthermore, the project focuses on nonparametric functional regression, with an emphasis on designing minimax optimal estimators using infinite-dimensional Stein's identities. The study also examines the trade-offs between statistics and computation in all the aforementioned methods. The common thread weaving through these endeavors is the synergy between various versions of Stein's identities and reproducing kernels, contributing substantially to the advancement of models, methods, and theories in contemporary nonparametric statistics.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.