Count distribution whose variance exceeds its mean, obtained by mixing a Poisson law with a gamma law, and which describes claim frequencies far better than pure Poisson.
The Poisson law forces variance to equal mean. Real claim frequencies almost never respect that constraint: their variance is larger, a phenomenon called overdispersion. Three 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 itself 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 distribution, and the gamma parameter measures heterogeneity. The practical consequence is twofold: ignoring overdispersion understates aggregate loss in the tail, and in a generalized linear model it makes coefficient standard errors too small, so rating variables get declared significant when they are not. The fix is either an explicit negative binomial law or an estimated dispersion parameter.
Motor book of 180,000 policies, 2025 year. Observed mean frequency 5.1%, observed variance of claim count per policy 0.068 against 0.051 expected under Poisson, a dispersion ratio of 1.33. Under negative binomial the confidence interval on the driver age coefficient widens by 15%, and two territory levels that looked significant under Poisson stop being so.
surdispersion, overdispersion, loi de Poisson mélangée, Poisson-gamma