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 give equal importance to the forecasts at different locations regardless of differences in the prediction uncertainty. This is a useful feature when computing average scores but can be an unnecessarily strict requirement when mostly concerned with extremes. We propose the concept of local weight-scale invariance, describing scoring rules fulfilling local scale invariance in a certain region of interest, and as a special case local tail-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. The score is a suitable alternative 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, annual maximum rainfalls, and in an application to non-extreme forecast for the prediction of air pollution.

翻译：极端值统计分析可用于预测未来极端事件（如暴雨或破坏性风暴）的发生概率。这些预报的质量可通过评分规则进行衡量。局部尺度不变评分规则能对位于不同位置、具有不同预测不确定性的预报赋予同等重要性，这在计算平均得分时颇具优势，但当主要关注极端值时可能成为不必要的严格约束。我们提出局部权重-尺度不变性概念，描述在特定感兴趣区域内满足局部尺度不变性的评分规则，并针对大事件提出其特例——局部尾部尺度不变性。此外，我们开发并研究了一种新型加权连续分级概率评分（wCRPS）——缩放型wCRPS（swCRPS），该评分具有上述性质。该评分适用于在极端事件尺度存在空间差异时评估极端值模型，我们推导了广义极值分布下该评分的显式公式。通过模拟比较不同评分规则，并在极端水位、年最大降水量建模及非极端空气污染预报应用案例中展示其实际用法。