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：极值统计分析可用于预测未来极端事件（如强降雨或毁灭性风暴）的发生概率。此类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对不同位置的预报赋予相同权重，无需考虑预测不确定性的差异。这一特性在计算平均分数时具有实用价值，但当重点关注极端事件时可能构成不必要的严格约束。我们提出局部权重-尺度不变性概念，描述在特定关注区域内满足局部尺度不变性的评分规则，并针对极端事件提出其特例——局部尾尺度不变性。进一步地，我们开发并研究了一种具备该特性的加权连续排序概率评分（wCRPS）新版本——缩放版wCRPS（swCRPS）。该评分适用于对极端事件尺度存在空间差异的极值模型进行评分，并推导出广义极值分布下该评分的显式表达式。通过模拟对比验证各评分规则效果，并在极端水位、年最大降雨量建模以及非极端空气污染预报应用中展示其使用方法。