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

TL;DR

This paper extends the functional delta-method by introducing uniform quasi-Hadamard differentiability, enabling bootstrap consistency results for a broader class of functionals, with applications to risk measures and dependent data.

Contribution

It introduces the concept of uniform quasi-Hadamard differentiability, expanding the applicability of the bootstrap delta-method for a wider range of functionals.

Findings

Extended bootstrap delta-method to uniform quasi-Hadamard differentiability.

Proved a chain rule for composition of functionals.

Applied results to Average Value at Risk and dependent data bootstrap.

Abstract

The functional delta-method provides a convenient tool for deriving bootstrap consistency of a sequence of plug-in estimators w.r.t. a given functional from bootstrap consistency of the underlying sequence of estimators. It has recently been shown in Beutner and Z\"ahle (2016) that the range of applications of the functional delta-method for establishing bootstrap consistency in probability of the sequence of plug-in estimators can be considerably enlarged by replacing the usual condition of Hadamard differentiability of the given functional by the weaker condition of quasi-Hadamard differentiability. Here we introduce the notion of uniform quasi-Hadamard differentiability and show that this notion extends the set of functionals for which almost sure bootstrap consistency of the corresponding sequence of plug-in estimators can be obtained by the functional delta-method. We illustrate the benefit of our results by means of the Average Value at Risk functional as well as the composition of the Average Value at Risk functional and the compound convolution functional. For the latter we use a chain rule to be proved here. In our examples we consider the weighted exchangeable bootstrap for independent observations and the blockwise bootstrap for $\beta$-mixing observations.