In various applications of heavy-tail modelling, the assumed Pareto behaviour is tempered ultimately in the range of the largest data. In insurance applications, claim payments are influenced by claim management and claims may, for instance, be subject to a higher level of inspection at highest damage levels leading to weaker tails than apparent from modal claims. Generalizing earlier results of Meerschaert et al. (2012) and Raschke (2020), in this paper we consider tempering of a Pareto-type distribution with a general Weibull distribution in a peaks-over-threshold approach. This requires to modulate the tempering parameters as a function of the chosen threshold. Modelling such a tempering effect is important in order to avoid overestimation of risk measures such as the value-at-risk at high quantiles. We use a pseudo maximum likelihood approach to estimate the model parameters and consider the estimation of extreme quantiles. We derive basic asymptotic results for the estimators, give illustrations with simulation experiments and apply the developed techniques to fire and liability insurance data, providing insight into the relevance of the tempering component in heavy-tail modelling.

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使用威布尔分布的缓和帕累托型建模

在重尾建模的各种应用中，假设的帕累托行为最终在最大数据范围内得到缓和。在保险应用中，索赔支付受索赔管理的影响，例如，索赔可能会在最高损害级别接受更高级别的检查，导致尾部比模态索赔明显更弱。推广 Meerschaert 的早期结果等人. (2012) 和 Raschke (2020)，在本文中，我们考虑在峰值超过阈值的方法中对具有一般 Weibull 分布的帕累托型分布进行回火。这需要根据所选阈值来调节回火参数。为这种缓和效应建模很重要，以避免高估风险度量，例如高分位数的风险价值。我们使用伪最大似然方法来估计模型参数并考虑极端分位数的估计。我们为估计器推导出基本的渐近结果，通过模拟实验进行说明，并将开发的技术应用于火灾和责任保险数据，从而深入了解重尾建模中回火组件的相关性。