  It is informative to evaluate a forecaster's ability to predict outcomes that
have a large impact on the forecast user. Although weighted scoring rules have
become a well-established tool to achieve this, such scores have been studied
almost exclusively in the univariate case, with interest typically placed on
extreme events. However, a large impact may also result from events not
considered to be extreme from a statistical perspective: the interaction of
several moderate events could also generate a high impact. Compound weather
events provide a good example of this. To assess forecasts made for high-impact
events, this work extends existing results on weighted scoring rules by
introducing weighted multivariate scores. To do so, we utilise kernel scores.
We demonstrate that the threshold-weighted continuous ranked probability score
(twCRPS), arguably the most well-known weighted scoring rule, is a kernel
score. This result leads to a convenient representation of the twCRPS when the
forecast is an ensemble, and also permits a generalisation that can be employed
with alternative kernels, allowing us to introduce, for example, a
threshold-weighted energy score and threshold-weighted variogram score. To
illustrate the additional information that these weighted multivariate scoring
rules provide, results are presented for a case study in which the weighted
scores are used to evaluate daily precipitation accumulation forecasts, with
particular interest on events that could lead to flooding.
