Tweedie Distribution

For each $p \in (-\infty, 0] \cup [1, \infty)$, there is an EDM with variance function $V(\mu) = \mu^p$. The corresponding distribution is called the Tweedie distribution with power $p$, written $Y \sim \text{Tweedie}(\mu, \phi, p)$. The mean is $\mu$ and the variance is $\phi \mu^p$. Special cases identify standard families:

The gap $p \in (0, 1)$ is excluded: no Tweedie distribution exists for those powers because the corresponding cumulant function would not be convex on a non-degenerate parameter space.

The variance function $V(\mu) = \mu^p$ encodes how the spread of the data grows with the mean. Constant variance ($p = 0$) is Normal. Variance equal to the mean ($p = 1$) is Poisson. Variance proportional to the squared mean ($p = 2$) is Gamma. The whole continuum is achievable, with the surprising compound case appearing in $(1, 2)$.

The Tweedie family is closed under several operations relevant in GLM fitting: scaling, summing iid copies, and translating the mean via a link function. The MLE for a Tweedie GLM can be computed by IRLS with the variance function $\mu^p$ plugged into the standard machinery. Modern actuarial-pricing software (R tweedie, statmod, Python glm packages) implements exactly this.

The dispersion $\phi$ and power $p$ are both estimated from data in applied GLMs; identifying them simultaneously requires either a profile likelihood over $p$ or a saddlepoint approximation. With small samples the joint MLE can be unstable, particularly near the boundaries $p = 1$ (Poisson) and $p = 2$ (Gamma). Practical workflow: fix $p$ on a grid, fit the GLM for each, compare via profile log-likelihood.

For $p \in (1, 2)$, $\text{Tweedie}(\mu, \phi, p)$ is the distribution of $Y = \sum_{i=1}^N Z_i,$ where $N \sim \text{Poisson}(\lambda)$ and $Z_i$ are iid $\text{Gamma}(\alpha, \beta)$ independent of $N$, with parameters $\lambda = \frac{\mu^{2 - p}}{\phi (2 - p)}, \quad \alpha = \frac{2 - p}{p - 1}, \quad \beta = \frac{1}{\phi (p - 1) \mu^{p - 1}}.$ In particular, $P(Y = 0) = e^{-\lambda}$ is strictly positive, and conditional on $Y > 0$, $Y$ has a continuous density on $(0, \infty)$.

Think of an insurance portfolio in a one-year period. Each policyholder either has zero claims (with positive probability) or has a positive number of claims, each of positive Gamma-distributed size. The total annual claim amount $Y$ is exactly zero on policies with no claims and Gamma-sum-shaped on policies with at least one. The Tweedie distribution captures both regimes in a single parametric family.

This is why the Tweedie distribution is the workshop tool for any quantity that is exactly zero with positive probability and otherwise positive and right-skewed. Insurance claim severity by policy. Rainfall by day in a dry region. Health-care expenditure by individual in a year. The compound-Poisson-Gamma decomposition makes the model interpretable (claim frequency, claim size, both as separate parameters) while keeping the unified Tweedie GLM tractable.

The density on $(0, \infty)$ does not have a closed-form expression for general $p \in (1, 2)$ and must be computed by series expansion or saddlepoint approximation. Statistical software handles this; do not try to write the density yourself for production use.