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.
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: In practice, insurance claim data is almost always overdispersed. Default to negative binomial unless the data specifically shows equidispersion or underdispersion.
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.
NOTE: Both heterogeneity (varying rates across the population) and contagion (one event triggering another) produce overdispersion. The variance-to-mean ratio alone cannot distinguish between them, look for verbal cues in the problem.
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. The chi-squared test will show the largest contribution from the zero cell.
Bottom line
- Variance-to-mean = 1: Poisson. < 1: Binomial. > 1: Negative binomial
- 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 first. 30 seconds, eliminates two of three options. Then look for structural cues: "given at least one claim" = truncation. "Some policyholders never file" = zero-inflation. "Heterogeneous population" = NB. The structural cue often matters more than the ratio. "1-Under-Over" (ratio = 1 (Poisson), under 1 (binomial), over 1 (NB). "No cap, no binomial." "Truncation hides the truth") truncated ratio is lower than true.
The full lesson (about 1,157 words, 8 min read) adds 2 worked examples, all 5 common mistakes, a self-check, free in the app.
Learning objectives
- 2h
Browse all free Exam FAM lessons or jump into free Exam FAM practice questions.