Anscombe's tests of fit for the negative binomial distribution

The negative binomial model is an important and flexible two parameter distribution that models data from many application areas. Here we re-examine tests of fit for the negative binomial distribution that were introduced by Anscombe (1950); they are based on a dispersion statistic U and a third moment statistic T. Small sample power calculations are given for U and T. We are not aware that such powers have been given previously. We show Anscombe’s tests are smooth tests in the sense of Rayner and Best (1989). Comparisons are made with an empirical probability generating function test suggested by Meintanis (2005). We suggest U not be used and that decisions on the fit of data to the negative binomial be made using bootstrap p-values rather than comparison with standard errors as suggested by Anscombe (1950). We show that tests based on a fourth moment component of a smooth test statistic have good power.