The Likelihood Ratio Test for Poisson versus Binomial Distributions

The estimation of N in the binomial B(N, p) distribution is a considerably harder problem than the estimation of p. We approach it as a “boundary value” estimation and testing problem, where the boundary N = ∞ corresponds to a Poisson distribution for the data, whereas N < ∞ corresponds to a binomial distribution. The asymptotic distribution of the deviance statistic for testing the hypothesis that the true value of N is infinite is shown to be a 50–50 mixture between a point mass at zero and a chi-squared distribution. We show also that the asymptotic distribution is not a good approximation for small samples, discuss the application of the method, and compare it to alternative approaches.