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.
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.
"Berry–Esseen bounds for Chernoff-type nonstandard asymptotics in isotonic regression." Ann. Appl. Probab. 32 (2) 1459 - 1498, April 2022. https://doi.org/10.1214/21-AAP1716