基于复合模型和阈值选择方法的极端索赔建模
Modelling extreme claims via composite models and threshold selection methods
摘要：
非寿险投资组合中存在的巨额和极端索赔会影响（再）保险公司估计准备金的能力。超额过阈值方法提供了一种捕获和建模保险索赔数据的典型行为的方法。本文讨论了几种常用的整体分布的复合模型，结合一个两参数帕累托分布以上的阈值。我们已经探讨了几种阈值选择方法在估计储量时的表现，以及在不同的样本量和尾部属性下选择批量分布的效果。为了调查这一点，已经进行了模拟研究。我们的研究表明，当数据充足时，平方根规则在储量估计的质量方面具有整体最佳性能。第二个最好的是指数检验，特别是当数据的右尾是极端的时候。当样本量变小时，同时估计具有最好的性能。此外，批量分布的选择的影响似乎是相当大的，特别是当分布是重尾的。此外，它表明，$p_{\leq b}$的经验估计，索赔低于阈值的概率，是更强大的比理论。
英文摘要：
The existence of large and extreme claims of a non-life insurance portfolio influences the ability of (re)insurers to estimate the reserve. The excess over-threshold method provides a way to capture and model the typical behaviour of insurance claim data. This paper discusses several composite models with commonly used bulk distributions, combined with a 2-parameter Pareto distribution above the threshold. We have explored how several threshold selection methods perform when estimating the reserve as well as the effect of the choice of bulk distribution, with varying sample size and tail properties. To investigate this, a simulation study has been performed. Our study shows that when data are sufficient, the square root rule has the overall best performance in terms of the quality of the reserve estimate. The second best is the exponentiality test, especially when the right tail of the data is extreme. As the sample size becomes small, the simultaneous estimation has the best performance. Further, the influence of the choice of bulk distribution seems to be rather large, especially when the distribution is heavy-tailed. Moreover, it shows that the empirical estimate of $p_{\leq b}$, the probability that a claim is below the threshold, is more robust than the theoretical one.
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