Link Function

Study guides: Exam SRM, Exam MAS-I, Exam PCPA

A Link Function $g$ is a monotone, differentiable function in a Generalized Linear Model that connects the expected value of the response $\mu = E[Y]$ to the linear predictor $\eta = \mathbf{x}^\top\boldsymbol{\beta}$ :

$g(\mu) = \eta$

$= \beta_0 + \beta_1 x_1 + \cdots + \beta_p x_p$

$\mu = g^{-1}(\eta)$

The canonical link is the natural link arising from the exponential family form of the distribution; it often leads to simpler inference
The choice of link function ensures that fitted means $\hat{\mu}$ respect the support of $Y$ (e.g., positivity for counts and severities)

Common link functions:

Name	$g(\mu)$	$g^{-1}(\eta)$	Typical Use
Identity	$\mu$	$\eta$	Normal regression
Log	$\ln\mu$	$e^\eta$	Poisson (counts), Gamma (severity)
Logit	$\ln\!\frac{\mu}{1-\mu}$	$\frac{e^\eta}{1+e^\eta}$	Binomial (binary outcomes)
Reciprocal	$1/\mu$	$1/\eta$	Gamma (inverse)
Probit	$\Phi^{-1}(\mu)$	$\Phi(\eta)$	Binary outcomes

The log link is standard for insurance severity and frequency models because it guarantees positive fitted values and allows multiplicative interpretation: a unit increase in $x_j$ multiplies $\mu$ by $e^{\beta_j}$

Interpreting a Log-Link Coefficient A Gamma GLM with log link models claim severity. The coefficient on the indicator for "sports car" is $\hat{\beta} = 0.35$ . Interpret this coefficient.

Answer With a log link, $\ln(\mu_{\text{sports}}) - \ln(\mu_{\text{non-sports}}) = 0.35$ , so $\mu_{\text{sports}}/\mu_{\text{non-sports}} = e^{0.35} \approx 1.42$ . Sports cars are associated with expected claim severity that is approximately 42% higher than non-sports cars, holding all else equal.