Peaks-over-threshold analysis using the generalized Pareto distribution is widely applied in modelling tails of univariate random variables, but much information may be lost when complex extreme events are studied using univariate results. In this paper, we extend peaks-over-threshold analysis to extremes of functional data. Threshold exceedances defined using a functional $r$ are modelled by the generalized $r$-Pareto process, a functional generalization of the generalized Pareto distribution that covers the three classical regimes for the decay of tail probabilities, and that is the only possible continuous limit for $r$-exceedances of a properly rescaled process. We give construction rules, simulation algorithms and inference procedures for generalized $r$-Pareto processes, discuss model validation, and use the new methodology to study extreme European windstorms and heavy spatial rainfall.

翻译：使用通用Pareto分配法进行的峰值超过临界值分析被广泛应用于单体随机变数的模拟尾巴,但在使用单体结果对复杂的极端事件进行研究时,可能会丢失大量信息。在本文中,我们将峰值超过阈值分析扩大到功能数据的极端情况。使用通用美元-Pareto(Pareto)程序模拟了使用通用美元-Pareto(Pareto)计算的阈值超过值。Pareto(Pareto)程序是通用美元-Pareto(Pareto)程序的一个功能性概括性分布法,它覆盖了尾体易腐蚀的三个古典制度,并且是适当重新标定的流程中唯一可能的连续限制。我们为通用美元-Pareto(Pareto)进程规定了建筑规则、模拟算法和推断程序,讨论模型验证,并使用新方法研究极端的欧洲风暴和空间降雨量大。