negative binomial frequency
Suppose you tried a Poisson model for car claims and it kept under-predicting the bad years and over-predicting the quiet ones — the real counts swing more wildly than a Poisson allows. The reason is usually that drivers are not all the same: most are careful, a few are reckless, and you do not know in advance who is who. The negative binomial frequency model captures exactly this extra spread.
The cleanest way to see it: imagine each policyholder has their own Poisson rate lambda, but lambda itself varies from person to person according to a gamma distribution (the cautious have small lambda, the reckless large). Average over that hidden variation and the resulting count distribution is the negative binomial. Its key feature is that the variance is larger than the mean — it is over-dispersed — controlled by an extra shape parameter. As that parameter grows, the negative binomial approaches the Poisson; tune it down and the tail of high counts gets heavier.
In practice the negative binomial is the go-to upgrade whenever claim counts show more variability than a Poisson can explain, which is most real-world portfolios where risks are heterogeneous. It belongs to the (a,b,0) class (with a positive), so it slots straight into the Panjer recursion for aggregate losses. The honest caveat: over-dispersion in your data does not prove the negative binomial is 'true'; it is one plausible mechanism (unobserved heterogeneity), but contagion or clustering of claims can also inflate variance, and a good analyst checks the fit rather than assuming the story.
A motor book has an average of 0.2 claims per policy but the observed variance of counts is 0.5 — far above the mean, so a Poisson (which forces variance = mean = 0.2) fails. A negative binomial with mean 0.2 and the extra dispersion parameter chosen to match a variance of 0.5 fits the swing in bad and good years properly.
A Poisson whose rate varies across policies becomes a negative binomial — variance now exceeds the mean.
Over-dispersion does not uniquely identify the negative binomial. Unobserved heterogeneity is one cause; claim clustering or contagion is another. The model fits the symptom (variance above mean) but the underlying mechanism still deserves a check.