A longitudinal snalysis of the impact of distance driven on the probability of car accidents

Using telematics data, we study the relationship between claim frequency and distance driven through different models by observing smooth functions. We used Generalized Additive Models (GAM) for a Poisson distribution, and Generalized Additive Models for Location, Scale, and Shape (GAMLSS) that we generalize for panel count data. To correctly observe the relationship between distance driven and claim frequency, we show that a Poisson distribution with fixed effects should be used because it removes residual heterogeneity that was incorrectly captured by previous models based on GAM and GAMLSS theory. We show that an approximately linear relationship between distance driven and claim frequency can be derived. We argue that this approach can be used to compute the premium surcharge for additional kilometers the insured wants to drive, or as the basis to construct Pay-as-you-drive (PAYD) insurance for self-service vehicles. All models are illustrated using data from a major Canadian insurance company.