Quasi-Monte Carlo integration of characteristic functions and the rejection sampling method

Abstract

Quasi-Monte Carlo methods are, in general, superior to Monte Carlo methods for multidimensional integration. But this superiority may no longer exist, when the integrands are discontinuous or when the integrals are performed by importance sampling combined with the rejection sampling method. The purpose of this paper is to improve the performance of Quasi-Monte Carlo integration of characteristic functions and improve the rejection sampling method in the Quasi-Monte Carlo setting. In a rather general case, a method of smoothing characteristic functions is proposed. The characteristic function is replaced by a continuous one, which has the same integral value. This smoothing method is demonstrated on the rejection method. An extended smoothed version which is a generalization of the method by Moskowitz and Caflish [Mathl. Comput. Modelling 23 (1996) 37–54] is described. Numerical experiments show that the extended smoothed method is much more efficient than the standard Quasi-Monte Carlo and the standard rejection method when used with quasi-random sequences.