  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.
