A Chernoff-type distribution is a non-normal 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 appears as the distributional limit in many nonregular estimation problems, the accuracy of Chernoff-type approximations has been largely unknown. In this talk, I will discuss 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.
This talk is based on joint work with Qiyang Han.