Link function
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.
Nelder and Wedderburn, Journal of the Royal Statistical Society, Series A, 1972: the generalized linear model ties the mean to the variables through a transformation, the link function, and that transformation sets 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: each criterion applies a coefficient to a base premium. Positivity is guaranteed, the rating table reads easily, and it matches the business intuition that a risky territory multiplies the premium. Hence its status as the default, for frequency and severity alike. The logit link serves binary variables such as lapse or acceptance, and returns odds rather than amounts. The log link has one quiet consequence: the model is multiplicative, so it admits only the interactions that have been declared, 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.
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