We explore past and recent developments in rare-event probability estimation with a particular focus on a novel Monte Carlo technique Empirical Likelihood Maximization (ELM). This is a versatile method that involves sampling from a sequence of densit
ies using MCMC and maximizing an empirical likelihood. The quantity of interest, the probability of a given rare-event, is estimated by solving a convex optimization program related to likelihood maximization. Numerical experiments are performed using this new technique and benchmarks are given against existing robust algorithms and estimators.
The Cross Entropy method is a well-known adaptive importance sampling method for rare-event probability estimation, which requires estimating an optimal importance sampling density within a parametric class. In this article we estimate an optimal imp
ortance sampling density within a wider semiparametric class of distributions. We show that this semiparametric version of the Cross Entropy method frequently yields efficient estimators. We illustrate the excellent practical performance of the method with numerical experiments and show that for the problems we consider it typically outperforms alternative schemes by orders of magnitude.
