Improved Cross-Entropy Method for Estimation

The cross-entropy (CE) method is an adaptive importance sampling procedure that has been successfully applied to a diverse range of complicated simulation problems. However, recent research has shown that in some high-dimensional settings, the likelihood ratio degeneracy problem becomes severe and the importance sampling estimator obtained from the CE algorithm becomes unreliable. We consider a variation of the CE method whose performance does not deteriorate as the dimension of the problem increases. We then illustrate the algorithm via a high-dimensional estimation problem in risk management

Efficient rare-event simulation for the maximum of heavy-tailed random walks

Let $(X_n:n\geq 0)$ be a sequence of i.i.d. r.v.'s with negative mean. Set $S_0=0$ and define $S_n=X_1+... +X_n$. We propose an importance sampling algorithm to estimate the tail of $M=\max \{S_n:n\geq 0\}$ that is strongly efficient for both light and heavy-tailed increment distributions. Moreover, in the case of heavy-tailed increments and under additional technical assumptions, our estimator can be shown to have asymptotically vanishing relative variance in the sense that its coefficient of variation vanishes as the tail parameter increases. A key feature of our algorithm is that it is state-dependent. In the presence of light tails, our procedure leads to Siegmund's (1979) algorithm. The rigorous analysis of efficiency requires new Lyapunov-type inequalities that can be useful in the study of more general importance sampling algorithms.Comment: Published in at http://dx.doi.org/10.1214/07-AAP485 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Simulating Tail Probabilities in GI/GI.1 Queues and Insurance Risk Processes with Subexponentail Distributions

This paper deals with estimating small tail probabilities of thesteady-state waiting time in a GI/GI/1 queue with heavy-tailed (subexponential) service times. The problem of estimating infinite horizon ruin probabilities in insurance risk processes with heavy-tailed claims can be transformed into the same framework. It is well-known that naive simulation is ineffective for estimating small probabilities and special fast simulation techniques like importance sampling, multilevel splitting, etc., have to be used. Though there exists a vast amount of literature on the rare event simulation of queuing systems and networks with light-tailed distributions, previous fast simulation techniques for queues with subexponential service times have been confined to the M/GI/1 queue. The general approach is to use the Pollaczek-Khintchine transformation to convert the problem into that of estimating the tail distribution of a geometric sum of independent subexponential random variables. However, no such useful transformation exists when one goes from Poisson arrivals to general interarrival-time distributions. We describe and evaluate an approach that is based on directly simulating the random walk associated with the waiting-time process of the GI/GI/1 queue, using a change of measure called delayed subexponential twisting -an importance sampling idea recently developed and found useful in the context of M/GI/1 heavy-tailed simulations