Comparing Regenerative-Simulation-Based Estimators of the Distribution of the Hitting Time to a Rarely Visited Set

We consider the estimation of the distribution of the hitting time to a rarely visited set of states for a regenerative process. In a previous paper, we provided two estimators that exploited the weak convergence of the hitting time divided by its expectation to an exponential as the rare set becomes rarer. We now add three new estimators, based on a corrected exponential, a gamma, and a bootstrap approach, the last possibly providing less biased estimators when the rare set is only moderately rare. Numerical results illustrate that all of the estimators perform similarly. Although the paper focuses on estimating a distribution, the ideas can also be applied to estimate risk measures, such as a quantile or conditional tail expectation.