In this paper, we compare different spatial models for modelling the risk premium for water damage insurance on the level of the policyholder. We evaluate four models that take the spatial variability into account: (1) the Intrinsic Conditional Auto-Regressive (ICAR) model; (2) the Besag, York, Mollier (BYM) model; (3) the independent random effects model; and (4) a spatial spline model. The models are compared on a huge data set from the Norwegian insurance company Gjensidige containing seven million observations of policyholders during the period 2011–2018. While Bayesian methods are most frequently used for inference in Gaussian Markov Random Field models, we take a frequentist approach and estimate the model parameters using Laplace approximated restricted maximum likelihood. Using the R package mgcv, we compare the different models for claim frequency, claim size and combined in a risk premium model in a comprehensive cross-validation study. Practical measures such as the loss ratio lift, double lift and Gini index are used to compare performance. Finally, we also compare mgcv with INLA and show that for reasonable big data sets we get identical estimates at a much lower computational cost.

在本文中，我们在投保人层面比较了不同的空间模型来模拟水损害保险的风险溢价。我们评估了四个考虑空间可变性的模型：（1）内在条件自回归（ICAR）模型；(2) Besag, York, Mollier (BYM) 模型；(3) 独立随机效应模型；(4)空间样条模型。这些模型在挪威保险公司 Gjensidige 的庞大数据集上进行了比较，该数据集包含 2011-2018 年期间对保单持有人的 700 万个观察结果。虽然贝叶斯方法最常用于高斯马尔可夫随机场模型中的推理，但我们采用频率论方法并使用拉普拉斯近似受限最大似然估计模型参数。使用 R 包mgcv，我们在综合交叉验证研究中比较了索赔频率、索赔规模和风险溢价模型中的不同模型。使用损失率提升、双提升和基尼指数等实际措施来比较性能。最后，我们还将mgcv与INLA进行比较，并表明对于合理的大数据集，我们以低得多的计算成本获得相同的估计。