Gamma and Tweedie Regression

Some financial and insurance targets are strictly positive and right-skewed — a bond's loss-given-default severity, a claim size once a claim has actually occurred. Fit ordinary linear regression to these and the model can predict impossible negative amounts, and it mishandles the fact that big losses are rarer but far more spread out than small ones. Gamma regression is built for exactly this shape. A further complication shows up when the target is mostly zero but occasionally a large positive amount — total insurance claims per policy per year, most of which are $0 — where Tweedie regression handles the zero-mass and the skewed positive part in one unified model, instead of two glued together.

The analogy: rainfall in a single day versus rainfall in a season

Gamma regression is like modeling how much rain falls on a day it does rain — always some positive amount, occasionally a lot, rarely a huge amount, never negative, never exactly zero. Tweedie regression is like modeling total rainfall over a month, where some months have genuinely zero rain (dry spell, an exact point mass at 0) and the rest have a right-skewed positive amount when it does rain. Trying to fit one ordinary regression to "total monthly rainfall" ignores that zero isn't just "a small positive number" — it's a qualitatively different outcome (no rain at all happened), and Tweedie regression is built to represent that mixture directly.

The models

Gamma regression models a strictly positive target $y>0$ via a log link, same structural idea as Poisson regression:

$$\log(\mu) = \beta_0 + \beta_1 x_1 + \dots, \qquad \text{Var}(y) = \phi \mu^2$$

Plain English: predictions are always positive (guaranteed by the log link), and — unlike Poisson, where variance equals the mean — Gamma's variance grows with the square of the mean, matching the empirical pattern that bigger losses have proportionally bigger absolute variability, i.e. roughly constant coefficient of variation.

Tweedie regression generalizes this with a power parameter $p$ controlling the mean-variance relationship, $\text{Var}(y) = \phi \mu^p$. For $1 < p < 2$, the Tweedie distribution is a compound Poisson-Gamma: a Poisson-distributed number of Gamma-distributed positive amounts summed together, which produces exactly the "point mass at zero plus a skewed positive tail" shape:

$$y = \sum_{i=1}^{N} Z_i, \quad N \sim \text{Poisson}(\lambda), \quad Z_i \sim \text{Gamma}(\alpha, \beta)$$

Plain English: if zero claims occur ($N=0$), the sum is exactly zero; if one or more claims occur, their sizes sum to a positive, right-skewed amount — one model captures the whole mixture.

Worked example 1: gamma regression on loss severity

Loss severity given default is modeled as $\log(\mu) = 1.0 + 0.4 \cdot \text{seniority\_junior}$ (an indicator, 1 if junior debt, 0 if senior). For senior debt: $\mu = e^{1.0} \approx 2.72$ (in units of $10,000, say $27,200). For junior debt: $\mu = e^{1.4} \approx 4.06$, i.e. about $40,600. Junior debt's expected severity is $e^{0.4}\approx1.49$ times senior debt's — a 49% higher expected loss, consistent with junior debt's lower recovery priority, and both predictions stay strictly positive by construction.

Worked example 2: Tweedie on a mostly-zero claims target

A book of 1,000 policies has 850 with zero claims for the year and 150 with positive claims averaging $3,000 (right-skewed — mostly $500-$1,500, a few over $20,000). A Tweedie model with $p=1.5$ fits this mixture directly: its fitted mean for a given policy, $\hat\mu$, already reflects both "probability of any claim" and "size if a claim happens" combined into one number — e.g. \hat\mu = 0.15 \times \3{,}000 = $450$ expected loss per policy, without ever needing to first fit a separate "claim yes/no" classifier and then a separate severity model and multiply the two by hand, since Tweedie's compound-Poisson-Gamma structure does that combination internally.

The zero-vs-positive mass split visible here for small $\lambda$ is the same shape Tweedie regression captures for its target variable — a real chance of exactly zero, and otherwise a right-skewed positive spread.

What this means in practice

Gamma regression is the standard choice for modeling any strictly-positive, right-skewed continuous outcome in finance — loss severities, position sizes — anywhere ordinary least squares would risk predicting a nonsensical negative value. Tweedie regression is the workhorse of actuarial pricing (insurance premiums, portfolio credit losses with many zero-loss accounts) because it avoids a two-stage frequency-times-severity pipeline and fits the zero-inflated, skewed target in one coherent GLM.