Exact oracle inequality for a sharp adaptive kernel density estimator

TL;DR

This paper introduces an adaptive kernel density estimator with a new cross-validation method that is both asymptotically efficient and minimax-adaptive over Sobolev spaces, providing sharp theoretical guarantees.

Contribution

It proposes a novel cross-validation technique for adaptive kernel density estimation that achieves optimality and adaptivity in a one-dimensional setting.

Findings

Estimator is asymptotically MISE-efficient.

Achieves sharp minimax adaptivity over Sobolev spaces.

Provides a new concentration inequality proof avoiding chaining.

Abstract

In one-dimensional density estimation on i.i.d. observations we suggest an adaptive cross-validation technique for the selection of a kernel estimator. This estimator is both asymptotic MISE-efficient with respect to the monotone oracle, and sharp minimax-adaptive over the whole scale of Sobolev spaces with smoothness index greater than 1/2. The proof of the central concentration inequality avoids "chaining" and relies on an additive decomposition of the empirical processes involved.