Let $\pi\in \Pi(\mu,\nu)$ be a coupling between two probability measures $\mu$ and $\nu$ on a Polish space. In this article we propose and study a class of nonparametric measures of association between $\mu$ and $\nu$, which we call Wasserstein correlation coefficients. These coefficients are based on the Wasserstein distance between $\nu$ and the disintegration $\pi_{x_1}$ of $\pi$ with respect to the first coordinate. We also establish basic statistical properties of this new class of measures: we develop a statistical theory for strongly consistent estimators and determine their convergence rate in the case of compactly supported measures $\mu$ and $\nu$. Throughout our analysis we make use of the so-called adapted/bicausal Wasserstein distance, in particular we rely on results established in [Backhoff, Bartl, Beiglb\"ock, Wiesel. Estimating processes in adapted Wasserstein distance. 2020]. Our approach applies to probability laws on general Polish spaces.
翻译:让我们在\ Pi (\ mu,\ nu) $\ pie\ in\ Pi (\\ mu,\ nu) $( nu) 中, 在波兰空间的两种概率度量 $\ mu$ 和 $\ nu$( $ 美元) 之间混合。 在本条中,我们提出并研究一种非参数度量 $\ mu$( mu$) 和 $\ nu$( nu) 之间关联的等级, 我们称之为 瓦塞斯坦 相关系数。 这些系数基于美元和 美元 和 美元 美元 折成 $\ pí x_ 1 美元( 美元) 之间的瓦塞斯坦 距离。 我们还建立了这一新类别计量标准的基本统计属性 : 我们为非常一致的估算器制定统计理论, 并且确定在 严格支持的 $\ mu$ 和 $\ nu$\ nu$ ( $) 。 我们在整个分析中, 我们使用所谓的调整/ 瓦塞斯坦 的距离, 尤其是我们依靠 [Backhhhhhhoff, legn, we be s be s begleglegn legn leged in the proclegal posk procal roclegyclegal lacal level.