Assessing goodness of fit for Poisson and negative binomial models with low mean

Testing goodness of fit for a Poisson distribution is routine when the mean is sufficiently large; the scaled deviance G² or Pearson's X² statistic follow approximate chi-square distributions and perform the task well. When the mean is low, typically less than one, the approximations to chi-square distributions are poor. In this paper we explore the underlying reasons for this behaviour and present a practical resolution of the problem, in both single distribution and regression contexts. An extension to negative binomial models is also given. This research is motivated by a real example drawn from road accident modelling.