Rates of almost sure convergence of plug-in estimates for distortion risk measures

In this article, we consider plug-in estimates for distortion risk measures as for instance the Value-at-Risk, the Expected Shortfall or the Wang transform. We allow for fairly general estimates of the underlying unknown distribution function (beyond the classical empirical distribution function) to be plugged in the risk measure. We establish strong consistency of the estimates, we investigate the rate of almost sure convergence, and we study the small sample behavior by means of simulations.