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 by 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, allowing for inference in settings where parametric assumptions may not be justified.

翻译：在许多应用研究中，学者们关注处理或暴露对目标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的框架。针对二值处理，Tchetgen Tchetgen 与 Shpitser (2012) 基于目标参数的影响函数提出了直接与间接效应的有效估计量。这些估计量具有多重稳健性和渐近正态性等优良性质，同时允许干扰参数以低于根号n的速率收敛。然而，在涉及连续处理的情境中，若不施加强参数假设，这些基于影响函数的估计量难以直接应用。本研究采用核平滑方法，受 Tchetgen Tchetgen 与 Shpitser (2012) 基于影响函数的估计量启发，提出了一种适用于连续处理场景的估计量。我们所提出的方法采用交叉拟合技术，降低了对干扰函数的平滑性要求，并允许其以低于目标参数的速率进行估计。此外，与基于影响函数的估计量类似，我们提出的估计量具有多重稳健性与渐近正态性，可在参数假设可能不成立的情境下进行统计推断。