A goodness of fit test for the Poisson distribution based on the empirical generating function

Abstract

The generating function g(t) of the Poisson distribution with parameter [lambda] is the only generating function satisfying the differential equation g'(t) = [lambda]g(t). Denoting by gn(t) the empirical generating function of a random sample X1,..., Xn of size n drawn from a distribution concentrated on the nonnegative integers, we propose Tn = n[integral operator]01[n(t)- g'n(t)]2 dt as a goodness of fit statistic for the composite hypothesis that the distribution of Xi is Poisson. Using a parametric bootstrap to have a critical value, and estimating this in turn by Monte Carlo the resulting test is shown to be consistent against alternative distributions with finite expectation.Poisson distribution goodness of fit empirical generating function bootstrapping Monte Carlo samples