We study the estimation of causal estimand involving the joint distribution of treatment and control outcomes for a single unit. In typical causal inference settings, it is impossible to observe both outcomes simultaneously, which places our estimation within the domain of partial identification (PI). Pre-treatment covariates can substantially reduce estimation uncertainty by shrinking the partially identified set. Recent work has shown that covariate-assisted PI sets can be characterized through conditional optimal transport (COT) problems. However, finite-sample estimation of COT poses significant challenges, primarily because the COT functional is discontinuous under the weak topology, rendering the direct plug-in estimator inconsistent. To address this issue, existing literature relies on relaxations or indirect methods involving the estimation of non-parametric nuisance statistics. In this work, we demonstrate the continuity of the COT functional under a stronger topology induced by the adapted Wasserstein distance. Leveraging this result, we propose a direct, consistent, non-parametric estimator for COT value that avoids nuisance parameter estimation. We derive the convergence rate for our estimator and validate its effectiveness through comprehensive simulations, demonstrating its improved performance compared to existing approaches.

翻译：本研究探讨了针对单一处理单元中处理组与对照组结果联合分布的因果估计量估计问题。在典型的因果推断设定中，同时观测到两种结果是不可能的，这使得我们的估计属于部分识别（PI）的范畴。预处理协变量可以通过缩小部分识别集显著降低估计的不确定性。近期研究表明，协变量辅助的部分识别集可以通过条件最优传输（COT）问题来表征。然而，COT的有限样本估计面临重大挑战，主要是因为COT泛函在弱拓扑下具有不连续性，导致直接插入估计量不一致。为解决这一问题，现有文献依赖于涉及非参数干扰统计量估计的松弛方法或间接方法。在本研究中，我们证明了COT泛函在由自适应Wasserstein距离诱导的更强拓扑下具有连续性。基于这一结果，我们提出了一种直接、一致且非参数的COT值估计量，该估计量避免了干扰参数估计。我们推导了该估计量的收敛速率，并通过全面仿真验证了其有效性，证明了其相较于现有方法具有更优的性能。