Tweedie family
Family of distributions whose variance is proportional to a power of the mean, allowing direct modeling of pure premium despite the probability mass at zero from claim-free policies.
Tweedie, 1984, “An index which distinguishes between some important exponential families”: the family that bears his name answers a pricing actuary's problem. Modeling pure premium directly runs into the zeros, since 95% of policies have no claim and no classical continuous law represents a variable that is zero most of the time. Standard practice works around it by modeling frequency with Poisson and average cost with gamma, then multiplying. Tweedie does the job in one pass. Variance equals the dispersion parameter times the mean to the power p; for p strictly between one and two, the law is exactly a compound Poisson with gamma severities, with a probability mass at zero and a continuous positive density beyond, the very shape of pure premium. One model, hence simpler industrialization. What is lost is the decomposition: the model does not say whether a rise in pure premium comes from frequency or severity, though one calls for prevention and the other for claims handling.
Homeowners rating, 2026 year, 240,000 policies of which 91% are claim free. A Tweedie model with p estimated at 1.62 returns an average pure premium of 168 EUR and a deviance within 1% of the frequency-severity approach. The lost decomposition proved costly six months later: the 9% rise seen on one segment came entirely from average cost, that is from materials prices, and not from a frequency drift the prevention team had started to address.
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