Wilson-Hilferty-type approximation for Poisson Random Variable

Abstract

The possibility to apply non-linear transformation of the Wilson-Hilferty type to the Poisson random variable is investigated using an analytic approach, in order to improve the convergence towards the large-parameter Gaussian limit. Considering the already existing non-linear transformation for the Gamma distribution, the original Wilson-Hilferty and a higher-order transformations are showed and approximations for the Gamma function are illustrated. From this, an approximate algorithm which exploits standard Gaussian random numbers is established, in order to generate Poisson random numbers of any parameter m. The algorithm performance is also assessed. The convenience of this approximate approach it resides in the fact that simulations often could be computationally demanding so that optimized algorithms have to be adopted for the efficient extraction of Poisson random numbers with variable parameter m.