Robust estimator of the ruin probability in infinite time for heavy-tailed distributions

The probability of ruin of an insurance company is one of the main risk measures considered in risk theory, and the problems of its calculation and approximation have attracted much attention. Statistical estimations have been developed on the ruin probability in infinite time for insurance loses from heavy-tailed distributions. However, these estimation suffer heavily from under-coverage or have a robustness problem, particularly when losses are contaminated by large variations in the arrival of claims. We therefore need another method for estimating the probability of ruin in infinite time for heavy-tailed losses. This is why, in this paper, we introduce a robust estimator of the infinite-time probability of ruin for such distributions. Our methodology is based on extreme value theory, which offers adequate statistical results for such distributions. Our approach is based on a sensitive distribution known as the t-Hill estimator (t-score or score moment estimation) for the index of any tail distribution and introduced in [Fabián and Stehlík (2009)]. We establish their asymptotic normality, and through a simulation study, illustrate their behavior in terms of absolute bias and mean squared error. The simulation results clearly show that our estimators perform well and that they are fairly robust to outliers.