# Uniformly Efficient Importance Sampling for the Tail Distribution of Sums of Random Variables

- 2008

#### Abstract

Successful efficient rare-event simulation typically involves using importance sampling tailored to a specific rare event. However, in applications one may be interested in simultaneous estimation of many probabilities or even an entire distribution. In this paper, we address this issue in a simple but fundamental setting. Specifically, we consider the problem of efficient estimation of the probabilities P PS n ≥ naa for large n, for all a lying in an interval , where S n denotes the sum of n independent, identically distributed light-tailed random variables. Importance sampling based on exponential twisting is known to produce asymptotically efficient estimates when reduces to a single point. We show, however, that this procedure fails to be asymptotically efficient throughout when contains more than one point. We analyze the best performance that can be achieved using a discrete mixture of exponentially twisted distributions, and then present a method using a continuous mixture. We show that a continuous mixture of exponentially twisted probabilities and a discrete mixture with a sufficiently large number of components produce asymptotically efficient estimates for all a ∈ simultaneously. 1. Introduction. The use of importance sampling for efficient rare-event simulation has been studied extensively (see, e.g., Bucklew [4], Heidelberger [9], Juneja and Shahabuddin [11] for surveys). Application areas for rare-event simulation include communications networks where information loss and large delays are important rare events of interest (as in Chang et al. [6]), insurance risk where the probability of ruin is a critical performance measure (e.g., Asmussen [2]), credit risk models in finance where large losses are a primary concern (e.g., Glasserman and Li [8]), and reliability systems where system failure is a rare event of utmost importance (e.g., Shahabuddin [15]). Successful applications of importance sampling for rare-event simulation typically focus on the probability of a single rare event. As a way of demonstrating the effectiveness of an importance sampling technique, the probability of interest is often embedded in a sequence of probabilities decreasing to zero. The importance sampling technique is said to be asymptotically efficient or asymptotically optimal if the second moment of the associated estimator decreases at the fastest possible rate as the sequence of probabilities approaches zero. Importance sampling based on exponential twisting produces asymptotically efficient estimates of rare-event probabilities in a wide range of problems. As in the setting we consider here, asymptotic efficiency typically requires that the twisting parameter be tailored to a specific event. …

**DOI:**10.1287/moor.1070.0276