On robustness in risk theory

In this paper, we consider the robustness properties of some estimators in the context of a few natural problems in risk theory. They are the calculation of excess of loss premiums, stop-loss premiums in both individual and collective risk models, and the probability of ruin. For that purpose, we introduce the influence function (and its empirical equivalent, the sensitivity function) and we apply it to the non-parametric plug-in estimators of those quantities. We find that they all have unbounded influence function. In order to obtain estimators with bounded influence function, we consider parametric estimators. We note that the shape of the influence function for any function of the parameters is fixed by the method used for estimating them. We propose the use of minimum distance methods in order to obtain robust estimators. We compare one of them, the minimum Cramér–von Mises estimator, with the maximum likelihood estimator and the non-parametric plug-in estimator with various illustrations.