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Statistical analysis of extremes can be used to predict the probability offuture extreme events, such as large rainfalls or devastating windstorms. Thequality of these forecasts can be measured through scoring rules. Locally scaleinvariant scoring rules give equal importance to the forecasts at differentlocations regardless of differences in the prediction uncertainty. This is auseful feature when computing average scores but can be an unnecessarily strictrequirement when mostly concerned with extremes. We propose the concept oflocal weight-scale invariance, describing scoring rules fulfilling local scaleinvariance in a certain region of interest, and as a special case localtail-scale invariance, for large events. Moreover, a new version of theweighted Continuous Ranked Probability score (wCRPS) called the scaled wCRPS(swCRPS) that possesses this property is developed and studied. The score is asuitable alternative for scoring extreme value models over areas with varyingscale of extreme events, and we derive explicit formulas of the score for theGeneralised Extreme Value distribution. The scoring rules are compared throughsimulation, and their usage is illustrated in modelling of extreme waterlevels, annual maximum rainfalls, and in an application to non-extreme forecastfor the prediction of air pollution.

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