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

该奖项的全部或部分资金来自《2021年美国救援计划法案》(公法117-2)。在现代统计学中，非参数方法是分析多变量和高维数据的基本工具包。然而，许多标准的非参数方法都面临着两个关键挑战。首先，这些方法的性能通常对调整参数的多个主观选择很敏感。第二，为估算目的而开发的方法通常不能直接用于统计推断。该项目旨在系统地开发一种在自然形状约束下同时解决这两个关键问题的多维非参数方法的新范例。特别是，本项目中将要开发的形状约束方法不仅将完全自动化，无需特别调整，而且还具有同时具有最优估计和推理的优点。该项目将通过课程开发、为本科生和研究生提供研究指导，特别是那些来自代表不足的群体的学生，以及暑期项目，将研究与教育相结合。本项目将侧重于两类相辅相成的研究问题。第一类问题旨在了解一类非标准广义区块估计器的潜力，以及最大似然或最小二乘等标准方法的缺陷。第二类问题的目的是在几个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程序。这些问题的解决方案的共同基础在于最近由PI及其合著者发起的多维形状约束估计器的非标准分布特征的新兴研究领域。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.