  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.
