For both financial and environmental applications, tail distributions often correspond to extreme risks and an accurate modeling is mandatory. The peaks-over-threshold model is a classic way to model the exceedances over a high threshold with the generalized Pareto distribution. However, for some applications, the choice of a high threshold is challenging and the asymptotic conditions for using this model are not always satisfied. The class of extended generalized Pareto models can be used in this case. However, the existing extended model have either infinite or null density at the threshold, which is not consistent with tail modeling. In the present article, we propose new extensions of the generalized Pareto distribution for which the density at the threshold is positive and finite. The proposed extensions provide better estimate of the upper tail index for low thresholds than existing models. They are also appropriate for high thresholds because in that case, the extended models simplify to the generalize Pareto model. The performance and flexibility of the models are illustrated with the modeling of temperature exceeding a low threshold and non-zero precipitations recorded in Montreal. For non-zero precipitation, the very low threshold of 0 is used.

中文翻译：

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尾部估计的灵活扩展广义帕累托分布

对于金融和环境应用，尾部分布通常对应于极端风险，并且必须进行准确的建模。peaks-over-threshold 模型是使用广义 Pareto 分布对超过高阈值的超出进行建模的经典方法。然而，对于某些应用，高阈值的选择具有挑战性，并且并不总是满足使用该模型的渐近条件。在这种情况下可以使用扩展的广义帕累托模型类。然而，现有的扩展模型在阈值处具有无限或零密度，这与尾部建模不一致。在本文中，我们提出了广义帕累托分布的新扩展，其中阈值处的密度为正且有限。与现有模型相比，建议的扩展为低阈值提供了更好的上尾指数估计。它们也适用于高阈值，因为在这种情况下，扩展模型简化为泛化 Pareto 模型。模型的性能和灵活性通过在蒙特利尔记录的温度超过低阈值和非零降水的建模来说明。对于非零降水，使用非常低的阈值 0。