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


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 Sn ≥ na for large n, for all a lying in an interval , where Sn 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.

DOI: 10.1287/moor.1070.0276

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@article{Glasserman2008UniformlyEI, title={Uniformly Efficient Importance Sampling for the Tail Distribution of Sums of Random Variables}, author={Paul Glasserman and Sandeep Juneja}, journal={Math. Oper. Res.}, year={2008}, volume={33}, pages={36-50} }