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：我们研究函数型处理下的效应估计问题，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（其目标是实数的函数）形成对比。通过考虑一个灵活的标量-函数边际结构模型，我们采用权重修正核岭回归（WMKRR）进行估计。权重的构建是通过直接最小化由WMKRR估计量分解产生的均匀平衡误差，而非基于特定的处理选择模型进行估计。尽管在WMKRR下推导出的均匀平衡误差具有复杂结构，但借助表示定理，有限维凸优化算法可用于高效求解所提出的权重。所提出的WMKRR估计量可在无需对真实权重函数施加任何平滑性假设的情况下达到最优收敛速率。相应的实证性能通过模拟研究和实际数据应用得到验证。