On Arbitrarily Underdispersed Discrete Distributions

Abstract We survey a range of popular generalized count distributions, investigating which (if any) can be arbitrarily underdispersed, that is, its variance can be arbitrarily small compared to its mean. A philosophical implication is that some models failing this simple criterion should not be considered as “statistical models” according to McCullagh’s extendibility criterion. Four practical implications are also discussed: (i) functional independence of parameters, (ii) double generalized linear models, (iii) simulation of underdispersed counts, and (iv) severely underdispersed count regression. We suggest that all future generalizations of the Poisson distribution be tested against this key property.