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"... In the context of rare-event simulation, splitting and importance sampling (IS) are the primary approaches to make important rare events happen more frequently in a simulation and yet recover an unbiased estimator of the target performance measure, with much smaller variance than a straightforward M ..."

Abstract
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In the context of rare-event simulation, splitting and importance sampling (IS) are the primary approaches to make important rare events happen more frequently in a simulation and yet recover an unbiased estimator of the target performance measure, with much smaller variance than a straightforward Monte Carlo (MC) estimator. Randomized quasi-Monte Carlo (RQMC) is another class of methods for reducing the noise of simulation estimators, by sampling more evenly than with standard MC. It typically works well for simulations that depend mostly on very few random numbers. In splitting and IS, on the other hand, we often simulate Markov chains whose sample paths are a function of a long sequence of independent random numbers generated during the simulation. In this paper, we show that RQMC can be used jointly with splitting and/or IS to construct better estimators than those obtained by either of these methods alone. We do that in a setting where the goal is to estimate the probability of reaching B before reaching (or returning to) A when starting from a distinguished state not in B, where A and B are two disjoint subsets of the state space and B is very rarely reached. This problem has several practical applications. The paper is in fact a two-in-one: the first part provides a guided tour of splitting techniques, introducing along the way some improvements in the implementation of the multilevel splitting. At the end of the paper, we also give examples of situations where splitting is not effective. For these examples, we compare different ways of applying IS and combining it with RQMC.