Rare-Event Simulation Without Variance Reduction: an Extreme Value Theory Approach

In estimating probabilities of rare events, crude Monte Carlo (MC) simulation is inefficient which motivates the use of variance reduction techniques. However, these latter schemes rely heavily on delicate analyses of underlying simulation models, which are not always easy or even possible. We propose the use of extreme value analysis, in particular the peak-over-threshold (POT) method which is popularly employed for extremal estimation of real datasets, in the simulation setting. More specifically, we view crude MC samples as data to fit on a generalized Pareto distribution. We test this idea on several numerical examples. The results show that our POT estimator appears more accurate than crude MC and, while crude MC can easily give a trivial probability estimate 0, POT outputs a non-trivial estimate with a roughly correct magnitude. Therefore, in the absence of efficient variance reduction schemes, POT appears to offer potential benefits to enhance crude MC estimates.