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

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 S n ≥ na 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. However, in applications, one is often interested in estimating many probabilities or an entire distribution. For instance, portfolio credit risk management requires the estimation of the tail of the loss distribution for a range of loss thresholds. This is needed in measuring the amount of capital required to protect against typical ...

Strongly efficient estimators for light-tailed sums

“…This suggests computing P (S n > n ) via importance sampling, in which the importance distribution is exactly the one that is used in deriving the large deviations lower bound (namely in which the increments are iid with common increment distribution given by the "exponential twist" having mean ). When this optimal exponential twisting (OET) is used, it has been shown that the resulting importance sampling algorithm is "logarithmically e cient", in the sense that the logarithm of the number of simulation runs required to compute P (S n > n ) to a given relative accuracy is now o (n) in n. In fact, the number of simulation runs that is required to accurately 2 compute P (S n > n ) grows at the rate of n 1/2 (see, for instance, Theorem 1 of Glasserman and Juneja (2006), see also Bucklew (2004)). Hence, a very substantial e ciency improvement can be realized by utilizing the OET in computing such rare-event probabilities.…”

Efficient Simulation of Large Deviation Events for Sums of Random Vectors Using Saddle-Point Representations

“…For example, b n = 1 − n −ξ for any ξ > 0 satisfies (15). For each n, let g n denote the pdf of the form (13) with parameters α, a n and b n chosen as above.…”

“…random variables (see e.g., [23], for a queuing application; [16] for applications in credit risk modeling). This problem has been extensively studied in rare event simulation literature (see e.g., [5], [13], [15], [17], [25], [26]). Essentially, the literature exploits the fact that the zero variance importance sampling estimator for P (X n ∈ A), though unimplementable, has a Markovian representation.…”

Efficient importance sampling in ruin problems for multidimensional regularly varying random walks

“…First we consider a parametric family of importance sampling distributions based on mixtures-the precise form of which is given in Section 4. The mixture idea has also been used in the rare-event simulation literature for light-tailed systems; see, for instance, Sadowsky and Bucklew (1990) and, more recently, Glasserman and Juneja (2008). In the light-tailed environment, mixtures are often used for nonconvex rare events to protect the behavior of the likelihood ratio due to rogue sample paths that deviate from the most likely large deviations path.…”