Coordinate-wise transformation of probability distributions to achieve a Stein-type identity
Coordinate-wise transformation of probability distributions to achieve a Stein-type identity
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概率分布的坐标变换以实现斯坦因式恒等式
DOI：
10.1007/s41884-021-00051-9
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发表时间：
2021
期刊：
影响因子：
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通讯作者：
Tomonari Sei
中科院分区：
文献类型：
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作者：
Ushio Tanaka;Masami Saga;Junji Nakano;松井 宗也;Muneya Matsui;Tomonari Sei
结果表明，对于任何给定的具有正则性条件的多维概率分布，存在唯一的坐标变换，使得变换后的分布满足斯坦因型恒等式。存在的充分条件称为分布的共积性。该证明基于 Wasserstein 空间的完全测地线子集上的能量最小化问题。该结果被认为是 Sklar 关于 copula 定理的替代方案，并且也被解释为对角缩放定理的推广。 Stein型恒等式应用于多变量数据的评级问题。提供了分段均匀密度的数值程序。还讨论了一些开放性问题。
It is shown that for any given multi-dimensional probability distribution with regularity conditions, there exists a unique coordinate-wise transformation such that the transformed distribution satisfies a Stein-type identity. A sufficient condition for the existence is referred to as copositivity of distributions. The proof is based on an energy minimization problem over a totally geodesic subset of the Wasserstein space. The result is considered as an alternative to Sklar’s theorem regarding copulas, and is also interpreted as a generalization of a diagonal scaling theorem. The Stein-type identity is applied to a rating problem of multivariate data. A numerical procedure for piece-wise uniform densities is provided. Some open problems are also discussed.