CAREER: New Paradigms of Estimation and Inference in Constrained Nonparametric Models

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This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). Nonparametric methods are a basic toolkit for analyzing multivariate and high-dimensional data in modern statistics. However, many standard nonparametric methods are known to face two key challenges. First, the performance of these methods is usually sensitive to multiple subjective choices of tuning parameters. Second, the methods developed for the purpose of estimation typically cannot be directly used for statistical inference. This project aims to systematically develop a new paradigm of multi-dimensional nonparametric methods under natural shape constraints that simultaneously resolves these two critical issues. In particular, the shape-constrained methods to be developed in this project will not only be fully automated without ad-hoc tuning, but also enjoy simultaneous optimal estimation and inference merits. The project will integrate research with education through course development, research mentoring for undergraduate and graduate students, especially those from underrepresented groups, and summer programs. This project will focus on two complementary categories of research problems. Problems in the first category aim at understanding the potentials of a class of non-standard generalized block estimators, and the drawbacks of standard methods such as the maximum likelihood or least squares. Problems in the second category aim at developing fully automated inference procedures for several canonical local and global inference targets using the non-standard methods, in a few related models. The common ground for the solutions to these problems lies in an emerging research area of non-standard distributional characterizations of multi-dimensional shape-constrained estimators initiated recently by the PI and his coauthors.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.