Homeowners' claims: mean 0.8, variance 1.6. Group health: mean 2.1, variance 1.9. Workers' comp: mean 0.3, variance 0.3. Each dataset points to a different distribution. The variance-to-mean ratio narrows your choices immediately.
To identify distributions for a given application, check the support, the relationship between mean and variance, and how the data were observed.
HIGH-FREQUENCY: The variance-to-mean ratio test appears repeatedly, both directly and inside larger problems.
From 2g: for any (a,b,0) distribution, the ratio is .
Most insurance data is overdispersed. The negative binomial dominates actuarial frequency modeling.
KEY: Overdispersion makes negative binomial a candidate among the (a,b,0) families. It does not prove that family; check the support, zero pattern, and population model.
Different policyholders have different underlying rates. If each follows Poisson() but varies:
When the mixing distribution is gamma, the result is negative binomial. Other mixers also produce overdispersion but not (a,b,0) distributions.
Common mistakes
- Applying the ratio test to truncated data. ZT Poisson has ratio < 1 among non-zero values. Concluding "binomial" from truncated data is wrong.
- Confusing heterogeneity with contagion. Both give overdispersion, both point to NB. The verbal distinction matters. Scenario describes risk classes → heterogeneity. Scenario describes one event triggering another → contagion.
- Ignoring zero-inflation. If observed zeros far exceed the fitted model's prediction, a standard (a,b,0) model is inappropriate. Compare the observed proportion of zeros with the probability of zero under the proposed distribution.
Bottom line
- Within (a,b,0), variance-to-mean distinguishes Poisson, binomial, and negative binomial. Outside that class, the ratio alone does not identify a family.
- Overdispersion is common: heterogeneous populations inflate variance beyond Poisson
- Both heterogeneity and contagion produce overdispersion. The ratio cannot distinguish them
- Zero-inflation: excess zeros need a zero-modified or zero-inflated (a,b,1) model
Exam shortcut
Compute the variance-to-mean ratio, then inspect the model and observation scheme. A heterogeneous Poisson population is overdispersed, but only a gamma mixer guarantees negative binomial counts. Truncation or extra zeros also changes the comparison. Use dispersion to narrow the stated candidate families. Verify the support, zero probability, and any truncation before choosing.
The full lesson (about 1,215 words, 8 min read) adds 2 worked examples, all 5 common mistakes, a self-check, free in the app.
Learning objectives
- 2h
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