Estimator selection: a new method with applications to kernel density estimation

TL;DR

This paper discusses the calibration of estimator selection methods, reviews existing approaches emphasizing minimal penalty, and introduces a new bandwidth selection method for kernel density estimation that is data-driven.

Contribution

It presents a novel intermediate estimator selection method for kernel density estimation, combining ideas from existing approaches with theoretical guarantees.

Findings

The new method is theoretically justified.

It provides a fully data-driven bandwidth selection strategy.

The approach bridges existing estimator selection techniques.

Abstract

Estimator selection has become a crucial issue in non parametric estimation. Two widely used methods are penalized empirical risk minimization (such as penalized log-likelihood estimation) or pairwise comparison (such as Lepski's method). Our aim in this paper is twofold. First we explain some general ideas about the calibration issue of estimator selection methods. We review some known results, putting the emphasis on the concept of minimal penalty which is helpful to design data-driven selection criteria. Secondly we present a new method for bandwidth selection within the framework of kernel density density estimation which is in some sense intermediate between these two main methods mentioned above. We provide some theoretical results which lead to some fully data-driven selection strategy.