THE DIRECTIONAL OPTIMAL TRANSPORT

• 成果类型: Article
• 署名作者: Nutz, Marcel; Wang, Ruodu
• 署名单位: Columbia University; University of Waterloo
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/21-AAP1712
• 发表日期: 2022
• 页码: 1400-1420
• 关键词: Bounds; inequalities; arbitrage; Duality

摘要

We introduce a constrained optimal transport problem where origins x can only be transported to destinations y >= x. Our statistical motivation is to describe the sharp upper bound for the variance of the treatment effect Y - X given marginals when the effect is monotone, or Y >= X. We thus focus on supermodular costs (or submodular rewards) and introduce a coupling P-* that is optimal for all such costs and yields the sharp bound. This coupling admits manifold characterizations-geometric, order-theoretic, as optimal transport, through the cdf, and via the transport kernel-that explain its structure and imply useful bounds. When the first marginal is atomless, P-*, is concentrated on the graphs of two maps which can be described in terms of the marginals, the second map arising due to the binding constraint.