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 calculat ion of excess of loss premiums, stop-loss premiums in both individual and c ollective risk models, and the probability of ruin. For that purpose, we in troduce the influence function (and its empirical equivalent, the sensitivi ty function) and we apply it to the non-parametric plug-in estimators of th ose quantities. We find that they all have unbounded influence function. In order to obtain estimators with bounded influence function, we consider pa rametric estimators. We note that the shape of the influence function for a ny function of the parameters is fixed by the method used for estimating th em. We propose the use of minimum distance methods in order to obtain robus t estimators. We compare one of them, the minimum Cramer-von Mises estimato r, with the maximum likelihood estimator and the non-parametric plug-in est imator with various illustrations. (C) 2001 Elsevier Science B.V. All right s reserved.