Bounded Relative Error Importance Sampling and Rare Event Simulation

We consider estimating tail events using exponential families of importance sampling distributions. When the cannonical sufficient statistic for the exponential family mimics the tail behaviour of the underlying cumulative distribution function, we can achieve bounded relative error for estimating tail probabilities. Examples of rare event simulation from various distributions including Tukey’s g&h distribution are provided.

Keywords: Rare event simulation, relative error, g&h distribution, Monte Carlo methods, Importance sampling, Cross-entropy, Rényi Divergence.