Sharpness of improved Fr'echet-Hoeffding bounds: an optimal transport approach
Sharpness of improved Fr'echet-Hoeffding bounds: an optimal transport approach
复制标题
改进的 Frechet-Hoeffding 边界的清晰度：一种最佳传输方法
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
A. Papapantoleon
中科院分区：
文献类型：
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作者：
Daniel Bartl;Michael Kupper;Thibaut Lux;A. Papapantoleon
改进的Fr'echet-Hoeffding界是对Fr'echet类概率分布的ad-hoc估计，它具有给定的边缘和在Rd的子集上的规定值.使用最佳的运输方法，我们首先提供了一个替代的推导改进的上限。为此，我们建立了一个对偶表示的约束运输问题的分布在上述Fr'echet类，并表明，改进的上Fr' echet-Hoeffding界属于允许的函数类的对偶优化问题。强对偶的证明是基于增凸泛函的一般表示结果和共轭的显式计算。我们进一步表明，改进的上Fr'echet-Hoeffding界是不是这个运输问题的对偶优化。反过来，我们证明了改进的上限是一个放松的版本的初始运输问题的对偶优化，从而证明锋利的放松Fr 'echet类的改进的上Fr'echet-Hoeffding界。这是通过直接构造某类目标函数的对偶优化器来实现的。
The improved Fr'echet--Hoeffding bounds are ad-hoc estimates on the Fr'echet class of probability distributions with given marginals and prescribed values on a subset of R d . Using an optimal transport approach, we first provide an alternative derivation of the improved upper bound. To this end, we establish a dual representation of a constrained transport problem over distributions in the aforementioned Fr'echet class and show that the improved upper Fr'echet--Hoeffding bound belongs to the class of admissible functions for the dual optimization problem. The proof of strong duality is based on a general representation result for increasing convex functionals and the explicit computation of the conjugates. We show further that the improved upper Fr'echet--Hoeffding bound is not the dual optimizer of this transport problem. In turn we prove that the improved upper bound is the dual optimizer of a relaxed version of the initial transport problem, thus proving sharpness of the improved upper Fr'echet--Hoeffding bound for the relaxed Fr'echet class. This is achieved by direct construction of the dual optimizers for a certain class of objective functions.