In extreme value theory and other related risk analysis fields, probability weighted moments (PWM) have been frequently used to estimate the parameters of classical extreme value distributions. This method-of-moment technique can be applied when second moments are finite, a reasonable assumption in many environmental domains like climatological and hydrological studies. Three advantages of PWM estimators can be put forward: their simple interpretations, their rapid numerical implementation and their close connection to the well-studied class of U-statistics. Concerning the later, this connection leads to precise asymptotic properties, but non asymptotic bounds have been lacking when off-the-shelf techniques (Chernoff method) cannot be applied, as exponential moment assumptions become unrealistic in many extreme value settings. In addition, large values analysis is not immune to the undesirable effect of outliers, for example, defective readings in satellite measurements or possible anomalies in climate model runs. Recently, the treatment of outliers has sparked some interest in extreme value theory, but results about finite sample bounds in a robust extreme value theory context are yet to be found, in particular for PWMs or tail index estimators. In this work, we propose a new class of robust PWM estimators, inspired by the median-of-means framework of Devroye et al. (2016). This class of robust estimators is shown to satisfy a sub-Gaussian inequality when the assumption of finite second moments holds. Such non asymptotic bounds are also derived under the general contamination model. Our main proposition confirms theoretically a trade-off between efficiency and robustness. Our simulation study indicates that, while classical estimators of PWMs can be highly sensitive to outliers.

翻译：在极值理论及相关风险分析领域，概率加权矩（PWM）被广泛用于估计经典极值分布参数。当二阶矩有限时（这一假设在气候学和水文学等环境研究中较为合理），该矩估计方法即可适用。PWM估计量具有三个优势：直观的统计解释、快速的数值实现，以及与已被深入研究的U统计量类的密切关联。这种关联虽能导出精确的渐近性质，但当指数矩假设在诸多极值场景下不成立时，常规技术（如Chernoff方法）无法应用，导致非渐近界的研究长期缺失。此外，大值分析也易受异常值的不利影响——例如卫星测量中的缺陷读数或气候模型运行中的潜在异常。近年来，极值理论领域开始关注异常值处理问题，但在稳健极值理论框架下关于有限样本界的研究仍属空白，特别是针对PWM或尾部指数估计量。本文借鉴Devroye等（2016）提出的均值中位数框架，构建了一类新型稳健PWM估计量。研究表明，在二阶矩有限的假设下，这类稳健估计量满足亚高斯不等式，并在一般污染模型下推导出相应的非渐近界。我们的核心结论从理论上证实了效率与稳健性之间的权衡关系。模拟实验表明，经典PWM估计量对异常值高度敏感。