Probabilistic bounded relative error for rare event simulation learning techniques

In rare event simulation, we look for estimators such that the relative accuracy of the output is “controlled” when the rarity is getting more and more critical. Different robustness properties of estimators have been defined in the literature. However, these properties are not adapted to estimators coming from a parametric family for which the optimal parameter is random due to a learning algorithm. These estimators have random accuracy. For this reason, we motivate in this paper the need to define probabilistic robustness properties. We especially focus on the so-called probabilistic bounded relative error property. We additionally provide sufficient conditions, both in general and Markov settings, to satisfy such a property, and hope that it will foster discussions and new works in the area.