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 λ is the only generating function satisfying the differential equation g′(t) = λg(t). Denoting by g$_n$(t) the empirical generating function of a random sample X$_1$,…, X$_n$ of size n drawn from a distribution concentrated on the nonnegative integers, we propose T$_n$ = n∫$_0$$^1$[Xg$_n$(t)− g'$_n$(t)]$^2$ dt as a goodness of fit statistic for the composite hypothesis that the distribution of X$_i$ 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.