BERRY-ESSEEN BOUNDS FOR CHERNOFF-TYPE NONSTANDARD ASYMPTOTICS IN ISOTONIC REGRESSION

• 成果类型: Article
• 署名作者: Han, Qiyang; Kato, Kengo
• 署名单位: Rutgers University System; Rutgers University New Brunswick; Cornell University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/21-AAP1716
• 发表日期: 2022
• 页码: 1459-1498
• 关键词: multivariate normal approximation; Concentration inequalities; steins method; nonnormal approximation; exchangeable pairs; limit distribution; monotone density; brownian-motion; CONVERGENCE; error

摘要

A Chernoff-type distribution is a nonnormal distribution defined by the slope at zero of the greatest convex minorant of a two-sided Brownian motion with a polynomial drift. While a Chernoff-type distribution is known to appear as the distributional limit in many nonregular statistical estimation problems, the accuracy of Chernoff-type approximations has remained largely unknown. In the present paper, we tackle this problem and derive Berry-Esseen bounds for Chernoff-type limit distributions in the canonical nonregular statistical estimation problem of isotonic (or monotone) regression. The derived Berry-Esseen bounds match those of the oracle local average estimator with optimal bandwidth in each scenario of possibly different Chernoff-type asymptotics, up to multiplicative logarithmic factors. Our method of proof differs from standard techniques on Berry-Esseen bounds, and relies on new localization techniques in isotonic regression and an anti-concentration inequality for the supremum of a Brownian motion with a Lipschitz drift.