A modified functional delta method and its application to the estimation of risk functionals (Q604360)

For a fixed distribution function \(g\) on the unit interval \([0,1]\) let the functional \(\rho_g\) be defined by \(\rho_g(F)=-\int_{-\infty}^\infty x\,dg(F(x))\) for all distribution functions \(F\) on the real line for which the integral is well-defined. If \(g\) (regarded as a measure) has compact support strictly inside the open interval \((0,1)\), then it is well-known that \(\rho_g\) is Hadamard differentiable at \(F\) with respect to the supremum norm. Hadamard differentiability may fail, however, if the support of \(g\) contains at least one the the boundary points \(0\) of \(1\). The authors introduce a notion of quasi-Hadamard differentiability, which is applicable in this case and works also with weighted sup-norms. It is shown that this concept is sufficiently strong to establish a functional delta method. The results are applied to several examples which are not covered by usual Hadamard differentiability.