Confidence Intervals for Randomized Quasi-Monte Carlo Estimators

International audienceRandomized Quasi-Monte Carlo (RQMC) methods provide unbiased estimators whose variance often converges at a faster rate than standard Monte Carlo as a function of the sample size. However, computing valid confidence intervals is challenging because the observations from a single randomization are dependent and the central limit theorem does not ordinarily apply. The natural solution is to replicate the RQMC process independently a small number of times to estimate the variance and use a standard confidence interval based on the normal or Student distribution. In this paper we investigate bootstrap methods for getting nonparametic confidence intervals for the mean using a modest number of replicates. Our main conclusion is that intervals based on the Student t distribution are more reliable than even the bootstrap t method on the integration problems arising from RQMC. In our examples, the individual RQMC estimates often had extreme kurtosis but always had mild skewness. That situation is very favorable for the standard confidence intervals. Simulation experiments cannot cover all the cases of interest, so we add some caveats in our conclusions