Towards zero variance estimators for rare event probabilities

TL;DR

This paper develops methods to improve importance sampling estimators for rare event probabilities by accurately approximating conditional densities, enabling more precise estimation in large deviation scenarios.

Contribution

It introduces a novel approach to approximate conditional densities for i.i.d. variables under rare event conditions, enhancing importance sampling techniques.

Findings

Provides algorithms for density approximation under rare events.

Demonstrates improved estimator accuracy through simulations.

Discusses the maximum feasible length of the approximations.

Abstract

Improving Importance Sampling estimators for rare event probabilities requires sharp approximations of conditional densities. This is achieved for events E_{n}:=(f(X_{1})+...+f(X_{n}))\inA_{n} where the summands are i.i.d. and E_{n} is a large or moderate deviation event. The approximation of the conditional density of the real r.v's X_{i} 's, for 1\leqi\leqk_{n} with repect to E_{n} on long runs, when k_{n}/n\to1, is handled. The maximal value of k compatible with a given accuracy is discussed; algorithms and simulated results are presented.