Tweedie Distribution

Study guides: Exam MAS-I, Exam PCPA

The Tweedie Distribution is the Exponential Family member whose variance is a power of the mean, $V(\mu) = \mu^{p}$ . For $1 < p < 2$ it is a compound Poisson–Gamma distribution: a Poisson number of Gamma-sized claims. That gives it a point mass at zero plus a right-skewed continuous part — exactly the shape of pure premium data, where most policies have no claim at all.

$\text{Var}(Y) = \phi\,\mu^{p}, \qquad 1 < p < 2$

$Y = \sum_{i=1}^{N} X_i, \quad N \sim \text{Poisson}(\lambda), \ X_i \sim \text{Gamma}$

$p$	Distribution	Response
$0$	Normal	Continuous, symmetric
$1$	Poisson	Counts
$1 < p < 2$	Compound Poisson–Gamma	Pure premium (mass at $0$ , skewed above)
$2$	Gamma	Severity
$3$	Inverse Gaussian	Heavy-tailed severity

$P(Y = 0) = e^{-\lambda} > 0$ : the Tweedie is the one member that puts genuine probability on exactly zero while remaining continuous elsewhere, which no Gamma or lognormal model can do
Fitted with a log link, it models pure premium in one step instead of fitting frequency and severity separately — convenient, but with $\phi$ and $p$ constant across records it implicitly assumes frequency and severity move in the same direction: a predictor that raises pure premium raises both components
$p$ is held constant across records. It may be estimated by the software along with $\boldsymbol\beta$ , chosen from candidate values by log-likelihood or cross-validation, or selected judgmentally (common choices $1.6$ , $1.67$ or $1.7$ ); values used in insurance modelling typically range from $1.5$ to $1.8$
The Dispersion Parameter $\phi$ is estimated from the data. The unit Deviance has a closed form for $1 < p < 2$ : $d(y, \mu) = 2\left[\dfrac{y^{2-p}}{(1-p)(2-p)} - \dfrac{y\,\mu^{1-p}}{1-p} + \dfrac{\mu^{2-p}}{2-p}\right]$
Separate frequency and severity models are more transparent and diagnose better; the Tweedie is preferred when the split is unavailable or when a single pure-premium score is what the pricing system consumes

Reading a Tweedie Pure Premium Model A Tweedie GLM ( $p = 1.6$ , log link) for pure premium gives $\hat\beta_0 = 5.30$ and a coefficient of $0.22$ for "prior claim". What pure premium does the model predict for a risk with a prior claim?

Answer $\ln\hat\mu = 5.30 + 0.22 = 5.52 \ \Longrightarrow\ \hat\mu = e^{5.52} = 249.6$ The base pure premium is $e^{5.30} = 200.3$ and the prior-claim relativity is $e^{0.22} = 1.246$ — a $24.6\%$ load, applied multiplicatively as in any log-link model.

Why Not Model Pure Premium with a Gamma? An actuary tries a Gamma GLM on pure premium and the fit fails outright. Why, and what does the Tweedie change?

Answer The Gamma density is defined only for $y > 0$ , but most policies have pure premium exactly $0$ — the likelihood is undefined for those rows. The Tweedie with $1 < p < 2$ assigns positive probability $e^{-\lambda}$ to $Y = 0$ and a continuous Gamma-like density above it, so the whole book can be fitted in a single model.