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.

翻译：多元概率分布预测在众多应用中不可或缺。评分规则能够评估预测准确性，并比较不同预测方法。我们提出了一个面向多元分布评分规则的理论框架，该框架涵盖了现有的二次评分和多元连续排序概率评分。我们展示了如何利用该框架生成新的评分规则。在某些多元情境中，实际需要的是水平集的预测——例如用于异常检测的密度水平集，或作为风险度量的累积分布水平集。这促使我们考虑针对此类水平集的评分函数。对于单变量分布，已有充分研究指出连续排序概率评分可表达为分位数评分的积分。我们证明，类似地，多元分布评分规则可被分解以获得水平集的评分函数。基于此，我们提出了针对不同类型水平集（包括密度水平集和累积分布水平集）的评分函数。为计算这些评分，我们设计了一种简单数值算法。我们通过模拟研究验证所提方法的有效性，并利用真实数据说明其在预测组合与CoVaR估计中的应用价值。