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

Eric Beutner, 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.

Original languageEnglish
Pages (from-to)1181-1222
Number of pages42
JournalElectronic Journal of Statistics
Issue number1
Publication statusPublished - 1 Jan 2016
Externally publishedYes


  • Bootstrap
  • Functional delta-method
  • Quasi-Hadamard differentiability
  • Statistical functional
  • Weak convergence for the open-ball σ-algebra

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