Bounded Relative Error Importance Sampling and Rare Event Simulation

We consider estimating tail events using exponential families of importance sampling distributions and show that under mild conditions on the exponential family we can achieve bounded relative error. Keywords: rare event simulation, relative error, g&h distribution, Monte Carlo methods, Importance sampling, Cross-entropy, Rényi Divergence. 1 Introduction Suppose X is a random variable with cumulative distribution function F and probability density function f with respect to Lebesgue measure. Suppose we wish to estimate an expected value such as E(g(X)) = Z 1 1 g(x)f(x)dx where g is an arbitrary integrable function. We wish to use importance sampling (IS): generateX from an alternative distribution in an exponential family having probability density f (x) = 1 m( ) e T f(x); (1) where m( ) = R e T f(x)dx. The modi…cation of the original density by the multiplication of a term like e T (x) when T (x) = x is variously referred to in the literature as an exponential twist or tilt of the density f but we will adopt this language to include a modi…cation of f to include a density of the form (1) and use the phrase standard exponential twist when T (x) = x: There are many potential Monte Carlo estimators of expected value which exploit speci…c features of the functions f and g to achieve variance reduction (see for example McLeish (2005), Chapter 4) but we will concentrate here on a speci…c technique,