Nonparametric estimation of risk measures of collective risks

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 in 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.