多变量分布的评分与水平集
Scores for Multivariate Distributions and Level Sets
AI总结：
本文提出了一种多变量分布评分规则的理论框架，用于生成针对密度和累积分布水平集的评分函数，并提出计算方法和应用实例。
摘要：
多变量概率分布的预测对于许多应用都是必不可少的。评分规则能够用于评估预测的准确性，并用于比较不同的预测方法。我们提出了一种针对多变量分布的评分规则的理论框架，该框架涵盖了现有的二次评分和多变量连续排名概率评分。我们展示了如何利用该框架生成新的评分规则。在某些多变量情境中，需要预测一个水平集，例如用于异常检测的密度水平集或用于风险衡量的累积分布水平集。这促使我们考虑这些水平集的评分函数。对于单变量分布，已知连续排名概率评分可以表示为量化评分的积分。我们展示了以类似方式，多变量分布的评分规则可以分解以获得水平集的评分函数。利用这一点，我们提出了不同类型的水平集的评分函数，包括密度水平集和累积分布水平集的评分函数。为了计算评分，我们提出了一种简单的数值算法。我们进行了一项模拟研究以支持我们的提议，并使用真实数据来说明其在预测结合和CoVaR估计中的实用性。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
