Asymptotic normality of plug-in level set estimates

TL;DR

This paper proves that the G-measure of the symmetric difference between a true level set and its kernel density plug-in estimator is asymptotically normally distributed, using Poissonization techniques.

Contribution

It establishes the asymptotic normality of the G-measure for level set estimation with kernel density estimators, demonstrating the effectiveness of Poissonization methods.

Findings

Asymptotic normality of the G-measure is proven.

Poissonization methods are effective in large sample theory.

The results facilitate statistical inference for level sets.

Abstract

We establish the asymptotic normality of the $G$-measure of the symmetric difference between the level set and a plug-in-type estimator of it formed by replacing the density in the definition of the level set by a kernel density estimator. Our proof will highlight the efficacy of Poissonization methods in the treatment of large sample theory problems of this kind.