Berry-Esseen bounds for Chernoff-type non-standard asymptotics in isotonic regression.

This paper derives Berry-Esseen bounds for an important class of non-standard asymptotics in nonparametric statistics with Chernoff-type limiting distributions, with a focus on the isotonic regression model. In the canonical situation of cube-root asymptotics, we obtain a cube-root Berry-Esseen bound (up to multiplicative logarithmic factors) for the speed of distributional approximation of the isotonic estimate to its Chernoff limit. Our method of proof relies on localization techniques in isotonic regression and an anti-concentration inequality for the supremum of a Brownian motion with a polynomial drift. These techniques extend to various Chernoff-type limiting distributions in isotonic regression with (i) general local smoothness conditions, (ii) both interior and boundary points, and (iii) general designs of covariates, where the Berry-Esseen bounds in each scenario match that of the oracle local average estimator with optimal bandwidth, up to multiplicative logarithmic factors.

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