We study treatment effect estimation with functional treatments where the average potential outcome functional is a function of functions, in contrast to continuous treatment effect estimation where the target is a function of real numbers. By considering a flexible scalar-on-function marginal structural model, a weight-modified kernel ridge regression (WMKRR) is adopted for estimation. The weights are constructed by directly minimizing the uniform balancing error resulting from a decomposition of the WMKRR estimator, instead of being estimated under a particular treatment selection model. Despite the complex structure of the uniform balancing error derived under WMKRR, finite-dimensional convex algorithms can be applied to efficiently solve for the proposed weights thanks to a representer theorem. The optimal convergence rate is shown to be attainable by the proposed WMKRR estimator without any smoothness assumption on the true weight function. Corresponding empirical performance is demonstrated by a simulation study and a real data application.

翻译：我们研究具有函数化处理的处理效应估计问题，其中平均潜在结果泛函是函数的函数，这与连续型处理效应估计（目标为实数的函数）形成对比。通过采用灵活的标量-函数边际结构模型，我们引入了一种权重修正核岭回归（WMKRR）方法进行估计。该方法的权重通过直接最小化由WMKRR估计量分解产生的均匀平衡误差来构建，而非基于特定处理选择模型进行估计。尽管WMKRR框架下导出的均匀平衡误差具有复杂结构，但由于表示定理的存在，我们可以应用有限维凸优化算法高效求解所提出的权重。研究表明，所提出的WMKRR估计量无需对真实权重函数施加光滑性假设即可达到最优收敛速度。通过模拟实验和真实数据应用验证了该方法相应的实证性能。