A New Approach to Sequential Stopping for Stochastic Simulation

In this paper, we solve the sequential stopping problem for a class of simulation problems in which variance estimation is difficult. We establish the asymptotic validity of sequential stopping procedures for estimators constructed using the sectioning (replication) methods with a fixed number of sections. The limiting distribution of the estimators at stopping times as the error size (the absolute error or the relative error) goes to 0 is characterized in closed form. This limiting distribution is different from the limiting distribution of the estimator constructed based on a fixed number of samples as the sample size goes to infinity, which indicates that we need a different scaling parameter when constructing the corresponding confidence intervals using the sequential stopping procedure. In particular, the scaling parameters we derived are larger than those suggested by the corresponding Student-t distribution. We also investigate the empirical performance of our proposed sequential stopping algorithms through some simulation experiments.