Transformation linking the predicted mean of a generalized linear model to the linear combination of variables, deciding whether a rate structure is additive, multiplicative, or logistic.
A generalized linear model does not model the mean directly, it models a transformation of the mean, and that transformation is the link function. The choice is not statistical but structural, because it determines the shape of the resulting rate. The identity link gives an additive rate where each criterion adds an amount, a structure rarely defensible since it permits a negative premium. The log link gives a multiplicative rate, where each criterion applies a coefficient to a base premium, which guarantees positivity, reads easily in a rating table, and matches business intuition: a risky territory multiplies, it does not add. That is why the log link is the default for both frequency and severity. The logit link serves binary variables such as lapse or acceptance, and returns odds rather than amounts. One quiet consequence deserves attention: under a log link the model is multiplicative, so it admits no interaction unless declared explicitly, and a real territory-by-age effect stays invisible until the cross term is added.
Motor frequency model, 2026 year, log link, 14 variables. The dense urban territory coefficient comes out at 0.262, an exponential factor of 1.30: frequency there is 30% higher than in the reference territory, all else equal. Under an identity link the same effect would have read as an addition of 1.4 frequency points, a value that would have driven the premium negative on the safest segments once combined with the other criteria.
link function, lien logarithmique, lien canonique, structure multiplicative du tarif