Exponential Family

Study guides: Exam SRM, Exam MAS-I

The Exponential Family is the class of distributions whose density can be written in a common exponential form. It is the set of response distributions a Generalized Linear Model can use: Normal, Poisson, binomial, Gamma, inverse Gaussian, and the Tweedie Distribution all belong to it.

$f(y \mid \theta, \phi) = \exp\!\left\{\frac{y\theta - b(\theta)}{a(\phi)} + c(y, \phi)\right\}$

$E[Y] = b'(\theta) = \mu, \qquad \text{Var}(Y) = a(\phi)\,b''(\theta) = \phi\,V(\mu)$

$\theta$ is the canonical parameter, $\phi$ the Dispersion Parameter, $b(\theta)$ the cumulant function whose derivatives generate the mean and variance
The variance function $V(\mu) = b''(\theta)$ is what distinguishes members and what makes the family so useful: it ties the spread to the mean instead of assuming it constant

Distribution	$V(\mu)$	Canonical link	Typical actuarial use
Normal	$1$	Identity	Ordinary Linear Regression
Poisson	$\mu$	$\ln\mu$	Claim frequency
Binomial	$\mu(1-\mu)$	$\ln\frac{\mu}{1-\mu}$	Retention, large-claim indicator
Gamma	$\mu^2$	$-1/\mu$ (log in practice)	Claim severity
Inverse Gaussian	$\mu^3$	$-1/(2\mu^2)$	Heavy-tailed severity
Tweedie ( $1<p<2$ )	$\mu^{p}$	$\mu^{1-p}/(1-p)$ (log in practice)	Pure premium

The canonical Link Function is the one with $g(\mu) = \theta$ ; it makes $\sum x_{ij}y_i$ a Sufficient Statistic and the likelihood well behaved, but any one-to-one, differentiable link may be used — the log link is standard for insurance because it makes rating factors multiplicative
Membership is what lets every GLM be fitted by the same algorithm — iteratively reweighted least squares, equivalent to Fisher scoring — with Fisher Information $X^\top W X/\phi$ ; the Deviance's $\chi^2$ distribution is a large-sample approximation. Quasi-likelihood runs the same algorithm from the mean–variance relationship alone, when a full exponential-family distribution is not assumed
Fixing $\phi$ at $1$ gives the "pure" one-parameter family; estimating it accommodates over- or under-dispersion without leaving the framework

Showing the Poisson Is in the Family Write the $\text{Poi}(\mu)$ probability function in exponential-family form and identify $\theta$ , $b(\theta)$ , and $V(\mu)$ .

Answer $f(y) = \frac{e^{-\mu}\mu^{y}}{y!} = \exp\left\{y\ln\mu - \mu - \ln y!\right\}$ Matching the standard form: $\theta = \ln\mu$ , $b(\theta) = e^{\theta} = \mu$ , $a(\phi) = 1$ , $c(y,\phi) = -\ln y!$ . $E[Y] = b'(\theta) = e^{\theta} = \mu, \qquad V(\mu) = b''(\theta) = e^{\theta} = \mu$ The canonical link is $\theta = \ln\mu$ — the log link, exactly what Poisson Regression uses.

Choosing a Member from the Data Plotting group variances against group means for a severity data set gives a relationship close to $\text{Var} \propto \mu^2$ . Which family member fits, and what would $\text{Var} \propto \mu$ have suggested?

Answer $V(\mu) = \mu^2$ is the Gamma, so a Gamma GLM (log link) is indicated for severity. A variance proportional to $\mu$ would point at the Poisson — appropriate for counts, not amounts. This mean–variance plot is the practical way to pick a distribution and is a standard Exploratory Data Analysis step before fitting.