Efficient rare event simulation for heavy-tailed systems via cross entropy

The cross entropy method is a popular technique that has been used in the context of rare event simulation in order to obtain a good selection (in the sense of variance performance tested empirically) of an importance sampling distribution. This iterative method requires the selection of a suitable parametric family to start with. The selection of the parametric family is very important for the successful application of the method. Two properties must be enforced in such a selection. First, subsequent updates of the parameters in the iterations must be easily computable and, second, the parametric family should be powerful enough to approximate, in some sense, the zero-variance importance sampling distribution. We obtain parametric families for which these two properties are satisfied for a large class of heavy-tailed systems including Pareto and Weibull tails. Our estimators are shown to be strongly efficient in these settings.