Sharpness of improved Fr\'echet-Hoeffding bounds: an optimal transport approach

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.