Locally tail-scale invariant scoring rules for evaluation of extreme value forecasts

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.