The widely used Poisson count process in insurance claims modeling is no longer valid if the claims occurrences exhibit dispersion. In this paper, we consider the aggregate discounted claims of an insurance risk portfolio under Weibull counting process to allow for dispersed datasets. A copula is used to define the dependence structure between the interwaiting time and its subsequent claims amount. We use a Monte Carlo simulation to compute the higher-order moments of the risk portfolio, the premiums and the value-at-risk based on the New Zealand catastrophe historical data. The simulation outcomes under the negative dependence parameter $\theta$, shows the highest value of moments when claims experience exhibit overdispersion. Conversely, the underdispersed scenario yields the highest value of moments when $\theta$ is positive. These results lead to higher premiums being charged and more capital requirements to be set aside to cope with unfavorable events borne by the insurers.

中文翻译：

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具有 Weibull 等待时间的 Copula 相关风险过程矩的蒙特卡罗模拟

如果索赔发生表现出离散，则保险索赔建模中广泛使用的泊松计数过程不再有效。在本文中，我们考虑在 Weibull 计数过程下保险风险组合的总折扣索赔，以允许分散的数据集。copula 用于定义间隔时间与其后续索赔金额之间的依赖关系结构。我们使用蒙特卡罗模拟来计算风险投资组合的高阶矩、保费和基于新西兰灾难历史数据的风险价值。负相关参数下的模拟结果$\theta$, 显示索赔经验表现出过度分散的时刻的最高值。相反，分散不足的场景产生最高值的时刻，当$\theta$是积极的。这些结果导致收取更高的保费，并留出更多的资本要求以应对保险公司承担的不利事件。