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

The functional delta-method provides a convenient tool for deriving the asymptotic distribution of plug-in estimators of statistical functionals from the weak convergence of the respective empirical processes. 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 can be considerably enlarged by employing the notion of quasi-Hadamard differentiability. Here we show that this enlargement carries over to the bootstrap. That is, for quasi-Hadamard differentiable functionals bootstrap consistency follows from bootstrap consistency of the respective empirical process. This enlargement requires weak convergence of the bootstrapped empirical process w.r.t.\ a nonuniform sup-norm. The latter is not problematic. Examples will be presented.