Array-RQMC to Speed up the Simulation for Estimating the Hitting-Time Distribution to a Rare Set of a Regenerative System

Estimating the distribution of the hitting time to a rarely visited set of states presents substantial challenges. We recently designed simulation-based esti­mators to exploit existing theory for regenerative systems that a scaled geometric sum of independent and identically distributed random variables weakly converges to an exponential random variable as the geometric’s parameter vanishes. The result­ing approximation then reduces the estimation of the distribution to estimating just the mean of the limiting exponential variable. The present work examines how ran­domized quasi-Monte Carlo (RQMC) techniques can help to reduce the variance of the estimators. Estimating hitting-time properties entails simulating a stochastic (here Markov) process, for which the so-called array-RQMC method is suited. After describing its application, we illustrate numerically the gain on a standard rare-event problem. This chapter combines ideas from several areas in which Pierre L’Ecuyer has made fundamental theoretical and methodological contributions: randomized quasi-Monte Carlo methods, rare-event simulation, and distribution estimation.