  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.
