Electronic Journal of Statistics

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

Eric Beutner and Henryk Zähle

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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.

Article information

Electron. J. Statist., Volume 10, Number 1 (2016), 1181-1222.

Received: October 2015
First available in Project Euclid: 5 May 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation 62G08: Nonparametric regression 62G20: Asymptotic properties 62G30: Order statistics; empirical distribution functions
Secondary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Bootstrap functional delta-method quasi- Hadamard differentiability statistical functional weak convergence for the open-ball $\sigma$-algebra


Beutner, Eric; Zähle, Henryk. Functional delta-method for the bootstrap of quasi-Hadamard differentiable functionals. Electron. J. Statist. 10 (2016), no. 1, 1181--1222. doi:10.1214/16-EJS1140. https://projecteuclid.org/euclid.ejs/1462450602

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