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 non-regular 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 non-regular 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.

翻译：切尔诺夫型分布是一种非正常的分布方式,由双面布朗运动的最大锥形微小点零点的斜坡以多元漂移来定义。虽然已知切尔诺夫型分布方式在许多非经常性统计估计问题中是分布极限,但切尔诺夫型近似的准确性仍然基本上未知。在本文件中,我们处理这一问题,并得出切尔诺夫型极限分布的Berry-Esseen界限,在异调(或单体内)回归的峡谷非常规统计估计问题中,产生Berry-Esseen 型分布。衍生的Berry-Esesearn 界限与甲骨文平均估量器的界限相匹配,在每一种可能不同的切诺夫型静脉图情景中都使用最佳的带宽度,最高可乘性对数系数。我们的证据方法不同于Berry-Eseseen边框的标准技术,在异调回归(或单体内)中依赖新的局部回归技术,并在布朗运动的顶部流流中采用抗浓缩不平等。