Improved Algorithms for Rare Event Simulation with Heavy Tails

The estimation of P(Sn > u) by simulation, where Sn is the sum of independent, identically distributed random varibles Y1, . . . , Yn, is of importance inmany applications. We propose two simulation estimators based upon the identity P(Sn > u) = nP(Sn > u, Mn = Yn), where Mn = max(Y1, . . . , Yn). One estimator uses importance sampling (for Yn only), and the other uses conditional Monte Carlo conditioning upon Y1, . . . , Yn−1. Properties of the relative error of the estimators are derived and a numerical study given in terms of the M/G/1 queue in which n is replaced by an independent geometric random variable N . The conclusion is that the new estimators compare extremely favorably with previous ones. In particular, the conditional Monte Carlo estimator is the first heavy-tailed example of an estimator with bounded relative error. Further improvements are obtained in the random-N case, by incorporating control variates and stratification techniques into the new estimation procedures.