Risk measures, distortion parameters, and their empirical estimation

Risk measures are of considerable current interest. Among other uses, they allow an insurer to calculate a risk-loaded premium for a random loss. However, the premium principle in use by the insurer may be, at least in part, based on considerations other than risk. It is then important to quantify the degree to which the premium compensates the insurer for the risk associated with the loss. This can be done by choosing a suitable risk measure and solving for the parameter that leads to the insurer's premium. When the loss distribution is unknown, this becomes a statistical estimation problem. 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.