Nonparametric estimation of risk measures of collective risks

Abstract We consider two nonparametric estimators for the risk measure of the sum of n i.i.d. individual insurance risks where the number of historical single claims that are used for the statistical estimation is of order n . This framework matches the situation that nonlife insurance companies are faced with within the scope of premium calculation. Indeed, the risk measure of the aggregate risk divided by n can be seen as a suitable premium for each of the individual risks. For both estimators divided by n we derive a sort of Marcinkiewicz–Zygmund strong law as well as a weak limit theorem. The behavior of the estimators for small to moderate n is studied by means of Monte-Carlo simulations.