Wasserstein 与 Cramér 距离的移位-离散分解
Shift-Dispersion Decompositions of Wasserstein and Cramér Distances
AI总结：
本文提出 Wasserstein 和 Cramér 距离的移位-离散分解，将分布差异分为有向移位与离散分量，增强可解释性，并应用于温度极值预报评估和经济学概率调查设计。
摘要：
散度函数是衡量概率分布之间距离或不相似性的度量，在统计学和应用中具有多种用途。我们提出了 Wasserstein 距离和 Cramér 距离的分解——这两种距离通过积分两个分布在分布函数或分位数函数上的差异来比较它们——分解为有方向的移位分量和离散分量。这些分量通过将分位数函数之间的差异分别划分为由移位和离散引起的贡献而获得。我们的分解以简洁的形式增加了关于分布如何不同的信息，从而增强了底层散度的可解释性。我们证明了我们的分解满足若干自然性质，并且在位置-尺度族中是唯一满足这些性质的。这些分解允许推导散度度量对位置和离散变化的敏感性，并产生了与通常的随机序和离散序相联系的弱随机序关系。我们的理论发展在两个应用中得到了说明，其中一个应用聚焦于温度极值的预报评估，另一个应用涉及经济学中概率调查的设计。
英文摘要：
Divergence functions are measures of distance or dissimilarity between probability distributions that serve various purposes in statistics and applications. We propose decompositions of Wasserstein and Cramér distances$-$which compare two distributions by integrating over their differences in distribution or quantile functions$-$into directed shift and dispersion components. These components are obtained by dividing the differences between the quantile functions into contributions arising from shift and dispersion, respectively. Our decompositions add information on how the distributions differ in a condensed form and consequently enhance the interpretability of the underlying divergences. We show that our decompositions satisfy a number of natural properties and are unique in doing so in location-scale families. The decompositions allow to derive sensitivities of the divergence measures to changes in location and dispersion, and they give rise to weak stochastic order relations that are linked to the usual stochastic and the dispersive order. Our theoretical developments are illustrated in two applications, where we focus on forecast evaluation of temperature extremes and on the design of probabilistic surveys in economics.
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