Approximation for general bootstrap of empirical processes with an application to kernel-type density estimation

TL;DR

This paper develops an approximation method for the generalized bootstrap of empirical processes, achieving optimal rates, and applies it to improve the understanding of bootstrapped kernel density estimators.

Contribution

It introduces a new approximation technique for the generalized bootstrap of empirical processes and applies it to kernel density estimation.

Findings

Achieves the rate in Kolmos et al. (1975) for the bootstrap approximation.

Provides an approximation for bootstrapped kernel density estimators.

Extends existing methods to generalized bootstrap scenarios.

Abstract

The purpose of this note is to provide an approximation for the generalized bootstrapped empirical process achieving the rate in Kolmos et al. (1975). The proof is based on much the same arguments as in Horvath et al. (2000). As a consequence, we establish an approximation of the bootstrapped kernel-type density estimator