Efficient Generation of Gaussian Varaiates Via Acceptance-Rejection Framework

Abstract

The Gaussian distribution is often considered to be the underlying distribution of many observed samples for modelling purposes, and hence simulation from the Gaussian density is required to verify the fitted model. Several methods, most importantly, Box-Muller method, inverse transformation method and acceptance-rejection method devised by Box and Muller¹, Rao et al.⁷ and Sigman⁸ respectively, are available in the literature to generate samples from the Gaussian distribution. Among these methods, Box-Muller method is the most popular and widely used because of its easy implementation and high efficiency,which produces exact samples². However, generalizing this method for generating non-standard multivariate Gaussian variates is not discovered yet. On the other hand, inverse transformation method uses numerical approximation to the CDF of Gaussian density which may not be desirable in some situations while performance of acceptance-rejection method depends on choosing efficient proposal density. In this paper, we introduce a more general technique by exploiting the idea invented by Wakefield⁹ under acceptance rejection framework to generate one dimensional Gaussian variates, in which we don’t require to choose any proposal density and it can be extended easily for non-standard multivariate Gaussian density. The proposed method is compared to the existing acceptance-rejection method (Sigman⁸ method), and we have shown both mathematically and empirically that the proposed method performs better than Sigman⁸ method as it has a higher acceptance rate (79.53 %) compared to Sigman (76.04 %) method.

Dhaka Univ. J. Sci. 67(2): 123-130, 2019 (July)