An M-estimator for rare-event probability estimation

Abstract

We describe a maximum-likelihood type estimator, or M-estimator, for Monte Carlo estimation of rare-event probabilities. In this method, we first sample from the zero-variance measure using Markov Chain Monte Carlo (MCMC), and then given the simulated data, we compute a maximum-likelihood-type estimator. We show that the resulting M-estimator is consistent, and that it subsumes as a special case the well-known fixed-effort splitting estimator. We give a numerical example of estimating accurately the tail distribution of the sum of log-normal random variables under a Gaussian copula. The numerical results suggests that for this example the method is competitive.