ESTIMATION AND INFERENCE FOR MINIMIZER AND MINIMUM OF CONVEX FUNCTIONS: OPTIMALITY, ADAPTIVITY AND UNCERTAINTY PRINCIPLES

• 成果类型: Article
• 署名作者: Cai, T. Tony; Chen, Ran; Zhu, Yuancheng
• 署名单位: University of Pennsylvania
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/24-AOS2355
• 发表日期: 2024
• 页码: 392-411
• 关键词: maximum; location

摘要

Optimal estimation and inference for both the minimizer and minimum of a convex regression function under the white noise and nonparametric regression models are studied in a nonasymptotic local minimax framework, where the performance of a procedure is evaluated at individual functions. Fully adaptive and computationally efficient algorithms are proposed and sharp minimax lower bounds are given for both the estimation accuracy and expected length of confidence intervals for the minimizer and minimum. The nonasymptotic local minimax framework brings out new phenomena in simultaneous estimation and inference for the minimizer and minimum. We establish a novel uncertainty principle that provides a fundamental limit on how well the minimizer and minimum can be estimated simultaneously for any convex regression function. A similar result holds for the expected length of the confidence intervals for the minimizer and minimum.