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We propose a Bayesian spline model which uses a natural cubicB-spline basis with knots placed at every development period to estimate the unpaid claims. Analogous to the smoothing parameter in a smoothing spline, shrinkage priors are assumed for the coefficients of basis functions. The accident period effect is modeled as a random effect, which facilitate the prediction in a new accident period. For model inference, we use Stan to implement the no-U-turn sampler, an automatically tuned Hamiltonian Monte Carlo. The proposed model is applied to the workers' compensation insurance data in the United States. The lower triangle data is used to validate the model.

Insurance companies have started to collect high-frequency GPS car driving data to analyze the driving styles of their policyholders. In previous work, we have introduced speed and acceleration heatmaps. These heatmaps were categorized with the K-means algorithm to differentiate varying driving styles. In many situations it is useful to have low-dimensional continuous representations instead of unordered categories. In the present work we use singular value decomposition and bottleneck neural networks (autoencoders) for principal component analysis. We show that a two-dimensional representation is sufficient to re-construct the heatmaps with high accuracy (measured by Kullback-Leibler divergences).

With the emergence of telematics car driving data, insurance companies have started to boost classical actuarial regression models for claim frequency prediction with telematics car driving information. In this paper, we propose two data-driven neural network approaches that process telematics car driving data to complement classical actuarial pricing with a driving behavior risk factor from telematics data. Our neural networks simultaneously accommodate feature engineering and regression modeling which allows us to integrate telematics car driving data in a one-step approach into the claim frequency regression models. We conclude from our numerical analysis that both classical actuarial risk factors and telematics car driving data are necessary to receive the best predictive models. This emphasizes that these two sources of information interact and complement each other.

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