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.

在重尾模型的各种应用中，假定的帕累托行为最终会在最大数据范围内调整。在保险申请中，理赔支付受到理赔管理的影响，例如，在最高损害水平下，理赔可能要接受更高级别的检查，从而导致比模态理赔明显的拖尾。概括Meerschaert等人的早期结果。（2012）和Raschke（2020），在本文中，我们考虑采用峰值-阈值方法对具有一般Weibull分布的帕累托型分布进行回火。这要求根据选择的阈值来调节回火参数。为了避免高估风险度量（例如高分位数的风险价值），对这种回火效应进行建模非常重要。我们使用伪最大似然方法来估计模型参数，并考虑对极端分位数的估计。我们为估算器得出基本的渐近结果，并通过仿真实验进行说明，并将开发的技术应用于火灾和责任保险数据，以深入了解回火组件在重尾模型中的相关性。