In many applications, researchers are interested in the direct and indirect causal effects of a treatment or exposure on an outcome of interest. Mediation analysis offers a rigorous framework for identifying and estimating these causal effects. For binary treatments, efficient estimators for the direct and indirect effects are presented in Tchetgen Tchetgen and Shpitser (2012) based on the influence function of the parameter of interest. These estimators possess desirable properties, such as multiple-robustness and asymptotic normality, while allowing for slower than root-n rates of convergence for the nuisance parameters. However, in settings involving continuous treatments, these influence function-based estimators are not readily applicable without making strong parametric assumptions. In this work, utilizing a kernel-smoothing approach, we propose an estimator suitable for settings with continuous treatments inspired by the influence function-based estimator of Tchetgen Tchetgen and Shpitser (2012). Our proposed approach employs cross-fitting, relaxing the smoothness requirements on the nuisance functions, and allowing them to be estimated at slower rates than the target parameter. Additionally, similar to influence function-based estimators, our proposed estimator is multiply robust and asymptotically normal, making it applicable for inference in settings where a parametric model cannot be assumed.

翻译：在许多应用中，研究者关注处理或暴露变量对感兴趣结果的直接与间接因果效应。中介分析为识别和估计这些因果效应提供了严谨框架。针对二元处理变量，Tchetgen Tchetgen 与 Shpitser (2012) 基于目标参数的 influential function 提出了直接效应与间接效应的高效估计量。这些估计量具备多重稳健性与渐近正态性等理想性质，同时允许干扰参数以慢于根号n的速率收敛。然而，在涉及连续处理变量的场景中，若不依赖强参数假设，此类基于 influence function 的估计量难以直接应用。本研究借鉴 Tchetgen Tchetgen 与 Shpitser (2012) 的 influence function 估计量，利用核平滑方法提出适用于连续处理场景的估计量。所提方法采用交叉拟合技术，放宽了对干扰函数平滑性的要求，允许其以慢于目标参数的速率进行估计。此外，与基于 influence function 的估计量类似，该估计量兼具多重稳健性与渐近正态性，适用于无法假定参数模型的推断场景。