An Expansion of the Probability Mass Function that may be used to Improve Poisson Approximations

Abstract

Random variables defined on the natural numbers may often be approximated by Poisson variables. Just as normal approximations may be improved by the Edgeworth expansion, Poisson approximations may be substantially improved by a similar expansion. Both of these expansions require one to approximate a generating function. This paper will derive a separate expansion for improving Poisson approximations to tail probabilities by approximating the probability mass function directly. This new expansion is often useful for points in the tails of the distribution. Examples will be presented to illustrate the usefulness and accuracy of this new expansion.