Negative binomial distribution

Greenwood and Yule showed in 1920, in the Journal of the Royal Statistical Society, that individual accident counts fit the Poisson law poorly and fit a mixture well, the one that yields the negative binomial distribution. The Poisson law forces variance to equal mean. Real claim frequencies almost never respect that constraint: their variance is larger, which is overdispersion. Several causes stack. The book is heterogeneous, each policyholder carrying its own risk intensity. Claims arrive in clusters, one storm generating a hundred notifications on the same day. And the intensity varies year to year with weather or the economy. Mixing a Poisson law with a gamma law over its intensity parameter yields exactly a negative binomial, and the gamma parameter measures heterogeneity. Ignoring overdispersion understates aggregate loss in the tail; in a generalized linear model it also makes coefficient standard errors too small, so rating variables get declared significant when they are not. The fix is an explicit negative binomial or an estimated dispersion parameter.