Accuracy estimation for quasi-Monte Carlo simulations

The conventional Monte Carlo approach to integration and simulation is a useful alternative to analytic or quadrature methods. It has been recognized through theory and practice that a variety of uniformly distributed sequences provide more accurate results than a purely pseudorandom sequence. The improvement in accuracy depends on the number of dimensions and the discrepancy of the sequence, which are known, and the variation of the function, which is often not known. Unlike pseudorandom methods, the accuracy of a quasirandom simulation cannot be estimated using the sample variance of the evaluations or by bootstrapping. The improvement in time-to-accuracy using quasirandom methods can be as large as several orders of magnitude, so even an empirical accuracy estimator is worth pursuing. In this paper, we discuss several methods for quasirandom empirical accuracy estimation and evaluate a modified empirical technique that appears to be useful.