Functional delta-method for the bootstrap of quasi-Hadamard differentiable functionals

TL;DR

This paper extends the functional delta-method to bootstrap procedures for quasi-Hadamard differentiable functionals, enabling broader applications by establishing bootstrap consistency under more general convergence conditions.

Contribution

It demonstrates that bootstrap consistency for quasi-Hadamard differentiable functionals follows from empirical process bootstrap consistency, expanding the method's applicability.

Findings

Bootstrap consistency follows from empirical process bootstrap consistency.

Enlarged application scope via nonuniform sup-norm convergence.

Illustrative examples demonstrate practical relevance.

Abstract

The functional delta-method provides a convenient tool for deriving the asymptotic distribution of a plug-in estimator of a statistical functional from the asymptotic distribution of the respective empirical process. Moreover, it provides a tool to derive bootstrap consistency for plug-in estimators from bootstrap consistency of empirical processes. It has recently been shown that the range of applications of the functional delta-method for the asymptotic distribution can be considerably enlarged by employing the notion of quasi-Hadamard differentiability. Here we show in a general setting that this enlargement carries over to the bootstrap. That is, for quasi-Hadamard differentiable functionals bootstrap consistency of the plug-in estimator follows from bootstrap consistency of the respective empirical process. This enlargement often requires convergence in distribution of the bootstrapped empirical process w.r.t.\ a nonuniform sup-norm. The latter is not problematic as will be illustrated by means of examples.