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Bayesian credibility under a bivariate prior on the frequency and the severity of claims
Insurance: Mathematics and Economics ( IF 1.9 ) Pub Date : 2021-06-24 , DOI: 10.1016/j.insmatheco.2021.06.003
Eric C.K. Cheung , Weihong Ni , Rosy Oh , Jae-Kyung Woo

In this paper, as opposed to the usual assumption of independence, we propose a credibility model in which the (unobservable) risk profiles of the claim frequency and the claim severity are dependent. Given the risk profiles, the (conditional) marginal distributions of frequency and severity are assumed to belong to the exponential family. A bivariate conjugate prior is proposed for the risk profiles, where the dependency is incorporated via a factorization structure of the joint density. The bivariate posterior is derived, and in turn the Bayesian premium for the aggregate claim is given along with some results on the predictive joint and marginal distributions involving the claim number and the aggregate claim in the next period. To demonstrate the generality of our proposed model, we provide four different examples of bivariate conjugate priors in relation to mixed Erlang, gamma mixture, Farlie-Gumbel-Morgenstern (FGM) copula, and bivariate beta, where each choice has different merits. In these examples, more explicit results can be obtained, and in particular the predictive variance and Value-at-Risk (VaR) of the aggregate claim certainly provide more information on the inherent risk than the Bayesian premium which is merely the predictive mean. Finally, numerical examples will be given to illustrate the effect of dependence on the results, including the use of a real data set that further takes observable risk factors into consideration under a regression setting.



中文翻译:

索赔频率和严重性的二元先验下的贝叶斯可信度

在本文中,与通常的独立性假设相反,我们提出了一个可信度模型,其中索赔频率和索赔严重性的(不可观察的)风险概况是相关的。给定风险概况,假设频率和严重性的(条件)边际分布属于指数族。为风险概况提出了双变量共轭先验,其中通过联合密度的分解结构合并依赖关系。推导出二元后验,然后给出总索赔的贝叶斯溢价以及涉及索赔数量和下一个时期总索赔的预测联合和边际分布的一些结果。为了证明我们提出的模型的普遍性,我们提供了与混合 Erlang、伽马混合、Farlie-Gumbel-Morgenstern (FGM) copula 和双变量 beta 相关的四个不同的双变量共轭先验示例,其中每个选择都有不同的优点。在这些例子中,可以获得更明确的结果,特别是总索赔的预测方差和风险价值 (VaR) 肯定比贝叶斯溢价提供更多的关于固有风险的信息,贝叶斯溢价只是预测平均值。最后,将给出数值例子来说明依赖对结果的影响,包括使用真实数据集,在回归设置下进一步考虑可观察到的风险因素。在这些例子中,可以获得更明确的结果,特别是总索赔的预测方差和风险价值 (VaR) 肯定比贝叶斯溢价提供更多的关于固有风险的信息,贝叶斯溢价只是预测平均值。最后,将给出数值例子来说明依赖对结果的影响,包括使用真实数据集,在回归设置下进一步考虑可观察到的风险因素。在这些例子中,可以获得更明确的结果,特别是总索赔的预测方差和风险价值 (VaR) 肯定比贝叶斯溢价提供更多的关于固有风险的信息,贝叶斯溢价只是预测平均值。最后,将给出数值例子来说明依赖对结果的影响,包括使用真实数据集,在回归设置下进一步考虑可观察到的风险因素。

更新日期:2021-06-30
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