On local $U$-statistic processes and the estimation of densities of functions of several sample variables

TL;DR

This paper introduces a local U-statistic process for density estimation of functions of multiple variables, establishing central limit theorems and inequalities that enhance understanding of its asymptotic behavior.

Contribution

It develops a new local U-statistic process framework and proves uniform central limit theorems, extending the theoretical foundation for density estimators of multivariate functions.

Findings

Established central limit theorems in various norms for the local U-statistic process.

Developed inequalities for U-processes applicable in broader contexts.

Provided uniform in bandwidth CLTs for density estimators of functions of several variables.

Abstract

A notion of local $U$-statistic process is introduced and central limit theorems in various norms are obtained for it. This involves the development of several inequalities for $U$-processes that may be useful in other contexts. This local $U$-statistic process is based on an estimator of the density of a function of several sample variables proposed by Frees [J. Amer. Statist. Assoc. 89 (1994) 517--525] and, as a consequence, uniform in bandwidth central limit theorems in the sup and in the $L_p$ norms are obtained for these estimators.