Simulating tail events with unspecified tail models

Reliable simulation estimation builds on accurately specified input models. In the context of simulating tail events, knowledge on the tail of the input model is especially important, yet is often hard to obtain due to a lack of data. In this paper, we consider tail event estimation without any knowledge on the input tail, but rather only making a general assumption that it is convex. We focus on the standard problem of estimating the probability for i.i.d. sum, and set out goal as to compute its worst-case bound among all summand distributions that have convex tails. Our main procedure relies on a stochastic, and in a sense infinite-dimensional, version of the Frank-Wolfe method in nonlinear programming. We demonstrate through a numerical example how the level of knowledge on the tail of the summands relates to the conservativeness in computing bounds for the aggregate tail quantity.