The acceptance-rejection method for low-discrepancy sequences

Abstract

Abstract Generation of pseudorandom numbers from different probability distributions has been studied extensively in the Monte Carlo simulation literature. Two standard generation techniques are the acceptance-rejection and inverse transformation methods. An alternative approach to Monte Carlo simulation is the quasi-Monte Carlo method, which uses low-discrepancy sequences, instead of pseudorandom numbers, in simulation. Low-discrepancy sequences from different distributions can be obtained by the inverse transformation method, just like for pseudorandom numbers. In this paper, we present an acceptance-rejection algorithm for low-discrepancy sequences. We prove a convergence result, and present error bounds. We then use this acceptance-rejection algorithm to develop quasi-Monte Carlo versions of some well-known algorithms to generate beta and gamma distributions, and investigate the efficiency of these algorithms numerically. We also consider the simulation of the variance gamma model, a model used in computational finance, where the generation of these probability distributions are needed. Our results show that the acceptance-rejection technique can result in significant improvements in computing time over the inverse transformation method in the context of low-discrepancy sequences.