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.

摘要

我们调查了一系列流行的广义计数分布，调查哪些（如果有的话）可以任意欠分散，也就是说，它的方差与其均值相比可以任意小。一个哲学含义是，根据 McCullagh 的可扩展性标准，一些不符合这个简单标准的模型不应被视为“统计模型”。还讨论了四个实际意义：(i) 参数的功能独立性，(ii) 双广义线性模型，(iii) 欠分散计数的模拟，以及 (iv) 严重欠分散计数回归。我们建议针对此关键属性测试泊松分布的所有未来推广。