| Summary: | Forecasts of multivariate probability distributions are required for a variety of applications. Scoring rules
enable the evaluation of forecast accuracy, and comparison between forecasting methods. We propose
a theoretical framework for scoring rules for multivariate distributions, which encompasses the existing
quadratic score and multivariate continuous ranked probability score. We demonstrate how this framework
can be used to generate new scoring rules. In some multivariate contexts, it is a forecast of a level set that
is needed, such as a density level set for anomaly detection or the level set of the cumulative distribution
as a measure of risk. This motivates consideration of scoring functions for such level sets. For univariate
distributions, it is well-established that the continuous ranked probability score can be expressed as the
integral over a quantile score. We show that, in a similar way, scoring rules for multivariate distributions can
be decomposed to obtain scoring functions for level sets. Using this, we present scoring functions for different
types of level set, including density level sets and level sets for cumulative distributions. To compute the
scores, we propose a simple numerical algorithm. We perform a simulation study to support our proposals,
and we use real data to illustrate usefulness for forecast combining and CoVaR estimation.
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