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

该奖项全部或部分由《2021年美国救援计划法案》（公法117-2）资助。非参数方法是现代统计学中分析多元高维数据的基本工具。然而，许多标准的非参数方法面临着两个关键的挑战。首先，这些方法的性能通常对调谐参数的多个主观选择很敏感。其次，为估计而开发的方法通常不能直接用于统计推断。本项目旨在系统地开发一种在自然形状约束下的多维非参数方法的新范式，同时解决这两个关键问题。特别地，在这个项目中开发的形状约束方法不仅可以完全自动化而不需要特别调优，而且可以同时具有最优估计和推理的优点。该项目将通过课程开发、对本科生和研究生（特别是来自代表性不足群体的学生）的研究指导以及暑期项目，将研究与教育结合起来。这个项目将集中于两类互补的研究问题。第一类问题旨在了解一类非标准广义块估计的潜力，以及诸如极大似然或最小二乘等标准方法的缺点。第二类问题的目的是在一些相关模型中使用非标准方法为几个规范的局部和全局推理目标开发全自动推理过程。这些问题的解决方案的共同点在于由PI和他的合作者最近发起的多维形状约束估计的非标准分布特征的新兴研究领域。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.