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April 2022 Berry–Esseen bounds for Chernoff-type nonstandard asymptotics in isotonic regression
Qiyang Han, Kengo Kato
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Ann. Appl. Probab. 32(2): 1459-1498 (April 2022). DOI: 10.1214/21-AAP1716


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.

Funding Statement

Q. Han was supported by NSF Grant DMS-1916221. K. Kato was supported by NSF Grants DMS-1952306 and DMS-2014636.


The authors would like to thank Jon Wellner for pointing out several references. They also would like to thank the anonymous referees, an Associate Editor and the Editor for their constructive comments that improve the quality of this paper.


Download Citation

Qiyang Han. Kengo Kato. "Berry–Esseen bounds for Chernoff-type nonstandard asymptotics in isotonic regression." Ann. Appl. Probab. 32 (2) 1459 - 1498, April 2022.


Received: 1 December 2019; Revised: 1 May 2021; Published: April 2022
First available in Project Euclid: 28 April 2022

Digital Object Identifier: 10.1214/21-AAP1716

Primary: 60E15 , 62G05

Keywords: anti-concentration , Berry–Esseen bound , Chernoff’s distribution , empirical process , Nonstandard asymptotics

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.32 • No. 2 • April 2022
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