Statistical analysis of extremes can be used to predict the probability of future extreme events, such as large rainfalls or devastating windstorms. The quality of these forecasts can be measured through scoring rules. Locally scale invariant scoring rules put equal importance on the forecasts at different locations regardless of differences in the prediction uncertainty. This can be an unnecessarily strict requirement when mostly concerned with extremes. We propose the concept of local tail-scale invariance, describing scoring rules fulfilling local scale invariance for large events. Moreover, a new version of the weighted Continuous Ranked Probability score (wCRPS) called the scaled wCRPS (swCRPS) that possesses this property is developed and studied. We show that the score is a suitable alternative to the wCRPS for scoring extreme value models over areas with varying scale of extreme events, and we derive explicit formulas of the score for the Generalised Extreme Value distribution. The scoring rules are compared through simulation, and their usage is illustrated in modelling of extreme water levels in the Great Lakes and annual maximum rainfalls in the Northeastern United States.

翻译：极端值的统计分析可用于预测未来极端事件（如大雨或破坏性风暴）的概率。这些预报的质量可通过评价准则进行衡量。局部尺度不变的评价准则对不同位置的预报赋予同等重要性，而不考虑预测不确定性的差异。当主要关注极端值时，这可能是过于严格的要求。我们提出了局部尾尺度不变性的概念，描述了针对大事件满足局部尺度不变性的评价准则。此外，开发并研究了一种具有该性质的新型加权连续排序概率评分（wCRPS）——即缩放版wCRPS（swCRPS）。研究表明，该评分是wCRPS在极端事件尺度变化区域对极端值模型进行评分的合适替代方案，并推导了广义极值分布下该评分的显式公式。通过模拟对比了这些评价准则，并在五大湖极端水位和美国东北部年最大降雨量建模中展示了其应用。