基于频率和烈度复合经验特征函数数值反演的总损失分布计算
Computing the aggregate loss distribution based on numerical inversion of the compound empirical characteristic function of frequency and severity
摘要：
本文提出了一种非参数的估算总损失分布（ALD）的方法，该方法通过结合和数值反演经验特征函数（CF）来实现。这种评估ALD的方法基于纯粹的非参数考虑，即，基于精算风险应用中索赔频率和严重程度的经验CF。然而，这种方法可以自然地推广到更复杂的半参数建模方法，例如，通过合并严重性分布重尾的广义帕累托分布拟合，和/或通过考虑参数CF（用于对专家知识建模）和经验CF（用于合并基于历史数据的知识-内部和/或外部）的加权混合。在这里，我们提出了一种简单而有效的方法和算法，用于CF的数值反演，适用于评估ALD和对应用重要的相关测量，例如，风险价值（VaR）。所提出的方法是基于组合的Gil-Pelaez反演公式推导的概率分布（PDF和CDF）从复合（经验）CF和梯形规则用于数值积分。所建议的方法的适用性说明了一个众所周知的保险数据集，丹麦火灾损失数据的分析。
英文摘要：
A non-parametric method for evaluation of the aggregate loss distribution (ALD) by combining and numerically inverting the empirical characteristic functions (CFs) is presented and illustrated. This approach to evaluate ALD is based on purely non-parametric considerations, i.e., based on the empirical CFs of frequency and severity of the claims in the actuarial risk applications. This approach can be, however, naturally generalized to a more complex semi-parametric modeling approach, e.g., by incorporating the generalized Pareto distribution fit of the severity distribution heavy tails, and/or by considering the weighted mixture of the parametric CFs (used to model the expert knowledge) and the empirical CFs (used to incorporate the knowledge based on the historical data - internal and/or external). Here we present a simple and yet efficient method and algorithms for numerical inversion of the CF, suitable for evaluation of the ALDs and the associated measures of interest important for applications, as, e.g., the value at risk (VaR). The presented approach is based on combination of the Gil-Pelaez inversion formulae for deriving the probability distribution (PDF and CDF) from the compound (empirical) CF and the trapezoidal rule used for numerical integration. The applicability of the suggested approach is illustrated by analysis of a well know insurance dataset, the Danish fire loss data.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
