Improving the rejection sampling method in quasi-Monte Carlo methods

The rejection sampling method is one of the most popular methods used in Monte Carlo methods. In this paper, we investigate and improve the performance of using a deterministic version of rejection method in quasi-Monte Carlo methods. It turns out that the “quality” of the point set generated by deterministic rejection method is closely related to the problem of quasi-Monte Carlo integration of characteristic functions, whose accuracy may be lost due to the discontinuity of the characteristic functions. We propose a method of smoothing characteristic functions in a rather general case. We replace the characteristic functions by continuous ones, without changing the value of the integrals. Using this smoothing technique, we modify the rejection method. An extended smoothed rejection method is described. Numerical experiments show that the extended smoothed rejection method is much more efficient than the standard quasi-Monte Carlo and the unsmoothed rejection method when used with low discrepancy sequences.