关于经验评分的合理性：Bonus Malus系统和一个新的Poisson混合模型
About the Justification of Experience Rating: Bonus Malus System and a new Poisson Mixture Model
摘要：
过去的索赔经验是计算保险合同公平价格的一个非常重要的信息。在许多欧洲国家，例如汽车保险的价格取决于司机在过去几年向保险公司报告的索赔数量。传统上，这些价格是根据具有伽马混合分布的混合泊松模型计算的。混合分布模型的汽车驾驶员的素质在保险的投资组合。这只是经验评级的一个例子。在经典的背景下，价格等于贝叶斯后验分布的期望。在某些业务领域（特别是第三方责任和暴露于极端天气事件的业务领域），我们认为经典的泊松-伽马模型无法很好地描述真实世界的数据。因此，我们研究的影响，混合分布的后验分布条件下的经验索赔的数量。这使得其他风险充足的保费原则比期望原则的应用。我们引入了一种新的混合分布--逆伽玛分布来模拟索赔数，并将其与经典伽玛分布进行了比较。在这两种情况下，可以找到混合分布的封闭解析表示：在经典情况下，众所周知的负二项分布，在我们的新情况下，使用贝塞尔函数表示。此外，我们给出了混合Poisson -逆- Gamma分布尾部行为的数值结果.最后，我们介绍了解决方案的概念。它使我们能够决定通过有经验的索赔数量对风险组进行分类是否是一个风险充分的程序。
英文摘要：
The claim experience of the past is a very important information to calculate the fair price of an insurance contract. In a lot of European countries for instance the prices for motor car insurance depend on the number of claims the driver has reported to the insurance company during the last years. Classically these prices are calculated on the basis of a mixed Poisson model with a gamma mixing distribution. The mixing distribution models the car drivers' qualities across the insured portfolio. This is just one example for experience rating. In the classical context the price is equal to the expectation of the Bayesian posterior distribution. In some lines of business (especially third party liability and lines with exposure to extreme weather events) we that the real world data cannot be described well enough by the classical Poisson - gamma model. Therefore we investigate the influence of the mixing distribution on the posterior distribution conditional on the experienced number of claims. This enables the application of other - more risk adequate premium principles than the expectation principle. We introduce the inverse - gamma distribution as a new mixing distribution to model claim numbers and compare it to the classical gamma distribution. In both cases a closed analytic representation of the mixed distribution can be found: In the classic case the well known negative binomial distribution, in our new one a representation using the Bessel functions. Additionally we present numerical results about the tail behaviour of the mixed Poisson - inverse - gamma distribution. Finally we introduce the concept of resolution. It enables us to decide if the classification of risk groups via the number of experienced claims is a risk adequate procedure.
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