Propagation of Input Tail Uncertainty in Rare-Event Estimation: A Light Versus Heavy Tail Dichotomy

We consider the estimation of small probabilities or other risk quantitiesassociated with rare but catastrophic events. In the model-based literature,much of the focus has been devoted to efficient Monte Carlo computation oranalytical approximation assuming the model is accurately specified. In thispaper, we study a distinct direction on the propagation of model uncertaintyand how it impacts the reliability of rare-event estimates. Specifically, weconsider the basic setup of the exceedance of i.i.d. sum, and investigate howthe lack of tail information of each input summand can affect the outputprobability. We argue that heavy-tailed problems are much more vulnerable toinput uncertainty than light-tailed problems, reasoned through their largedeviations behaviors and numerical evidence. We also investigate someapproaches to quantify model errors in this problem using a combination of thebootstrap and extreme value theory, showing some positive outcomes but alsouncovering some statistical challenges.