Understanding causal relationships in the presence of complex, structured data remains a central challenge in modern statistics and science in general. While traditional causal inference methods are well-suited for scalar outcomes, many scientific applications demand tools capable of handling functional data -- outcomes observed as functions over continuous domains such as time or space. Motivated by this need, we propose DR-FoS, a novel method for estimating the Functional Average Treatment Effect (FATE) in observational studies with functional outcomes. DR-FoS exhibits double robustness properties, ensuring consistent estimation of FATE even if either the outcome or the treatment assignment model is misspecified. By leveraging recent advances in functional data analysis and causal inference, we establish the asymptotic properties of the estimator, proving its convergence to a Gaussian process. This guarantees valid inference with simultaneous confidence bands across the entire functional domain. Through extensive simulations, we show that DR-FoS achieves robust performance under a wide range of model specifications. Finally, we illustrate the utility of DR-FoS in a real-world application, analyzing functional outcomes to uncover meaningful causal insights in the SHARE ({\em Survey of Health, Aging and Retirement in Europe}) dataset.

翻译：理解复杂结构化数据中的因果关系仍然是现代统计学乃至整个科学领域的核心挑战。虽然传统的因果推断方法非常适用于标量结果，但许多科学应用需要能够处理函数数据——即在连续域（如时间或空间）上观测到的函数结果——的工具。基于这一需求，我们提出了DR-FoS，一种用于在具有函数结果的观察性研究中估计函数平均处理效应（FATE）的新方法。DR-FoS展现出双重稳健特性，即使结果模型或处理分配模型之一存在误设，也能确保FATE的一致估计。通过利用函数数据分析和因果推断领域的最新进展，我们建立了该估计量的渐近性质，证明其收敛于一个高斯过程。这保证了在整个函数域上具有同步置信带的有效推断。通过大量模拟，我们表明DR-FoS在广泛的模型设定下均能实现稳健的性能。最后，我们通过一个真实世界应用展示了DR-FoS的实用性，通过分析函数结果来揭示SHARE（{\em 欧洲健康、老龄化和退休调查}）数据集中有意义的因果洞见。