Adaptive nonparametric estimation of a component density in a two-class mixture model

TL;DR

This paper introduces an adaptive nonparametric kernel estimator for the unknown component density in a two-class mixture model, utilizing a data-driven bandwidth selection method and providing theoretical guarantees and simulations.

Contribution

It proposes a novel randomly weighted kernel estimator with a fully data-driven bandwidth selection for the unknown component density in mixture models.

Findings

Oracle-type inequality for quadratic risk derived

Convergence rates established over Holder classes

Numerical simulations validate theoretical results

Abstract

A two-class mixture model, where the density of one of the components is known, is considered. We address the issue of the nonparametric adaptive estimation of the unknown probability density of the second component. We propose a randomly weighted kernel estimator with a fully data-driven bandwidth selection method, in the spirit of the Goldenshluger and Lepski method. An oracle-type inequality for the pointwise quadratic risk is derived as well as convergence rates over Holder smoothness classes. The theoretical results are illustrated by numerical simulations.