Bandwidth selection in kernel density estimation: Oracle inequalities and adaptive minimax optimality

TL;DR

This paper introduces a new kernel density estimator selection method that achieves minimax adaptivity over anisotropic Nikol'skii classes, supported by oracle inequalities and empirical process bounds.

Contribution

It develops a novel selection procedure for kernel estimators that ensures minimax optimality and adaptivity in density estimation under $ ext{L}_s$-loss.

Findings

The proposed method satisfies $ ext{L}_s$-risk oracle inequalities.

The estimator is minimax adaptive over anisotropic Nikol'skii classes.

Utilizes uniform bounds on empirical process $ ext{L}_s$-norms for derivations.

Abstract

We address the problem of density estimation with $\mathbb{L}_s$-loss by selection of kernel estimators. We develop a selection procedure and derive corresponding $\mathbb{L}_s$-risk oracle inequalities. It is shown that the proposed selection rule leads to the estimator being minimax adaptive over a scale of the anisotropic Nikol'skii classes. The main technical tools used in our derivations are uniform bounds on the $\mathbb{L}_s$-norms of empirical processes developed recently by Goldenshluger and Lepski [Ann. Probab. (2011), to appear].