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

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.