Nonparametric estimation of risk measures of collective risks

TL;DR

This paper introduces two nonparametric estimators for aggregate risk measures in insurance, analyzing their theoretical properties and performance for small to moderate sample sizes through simulations.

Contribution

It proposes novel nonparametric estimators for collective risk measures and establishes their asymptotic laws, with simulation studies on their finite-sample behavior.

Findings

Derived strong law of large numbers for the estimators

Established weak limit theorems for the estimators

Monte-Carlo simulations demonstrate estimator performance for small to moderate n

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 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.