Risk measures, distortion parameters, and their empirical estimation

In this paper, we investigate the nonparametric estimation of the parameter associated with a distortion-based risk measure. It is assumed that the premium principle is known, but no information is assumed about the loss distribution, and therefore empirical estimators are used. We explore the asymptotic properties of the resulting estimator of the risk measure parameter in general and for three well-known risk measures in particular: the proportional hazards transform, the Wang transform, and the conditional tail expectation.