Mediation analysis is crucial in many fields of science for understanding the mechanisms or processes through which an independent variable affects an outcome, thereby providing deeper insights into causal relationships and improving intervention strategies. Despite advances in analyzing the mediation effect with fixed/low-dimensional mediators and covariates, our understanding of estimation and inference of mediation functional in the presence of (ultra)-high-dimensional mediators and covariates is still limited. In this paper, we present an estimator for mediation functional in a high-dimensional setting that accommodates the interaction between covariates and treatment in generating mediators, as well as interactions between both covariates and treatment and mediators and treatment in generating the response. We demonstrate that our estimator is $\sqrt{n}$-consistent and asymptotically normal, thus enabling reliable inference on direct and indirect treatment effects with asymptotically valid confidence intervals. A key technical contribution of our work is to develop a multi-step debiasing technique, which may also be valuable in other statistical settings with similar structural complexities where accurate estimation depends on debiasing.

翻译：中介分析在众多科学领域中至关重要，它有助于理解自变量影响结果变量的机制或过程，从而为因果关系提供更深入的洞见并改进干预策略。尽管在固定/低维中介变量和协变量下的中介效应分析已取得进展，但在（超）高维中介变量和协变量存在时，我们对中介功能的估计与推断的理解仍然有限。本文提出了一种适用于高维设置的中介功能估计方法，该方法能够处理协变量与处理变量在生成中介变量时的交互作用，以及协变量与处理变量、中介变量与处理变量在生成响应变量时的交互作用。我们证明了所提出的估计量具有$\sqrt{n}$相合性与渐近正态性，从而能够基于渐近有效的置信区间对直接与间接处理效应进行可靠的统计推断。本研究的一项关键技术贡献是开发了一种多步去偏技术，该技术对于其他具有类似结构复杂性且精确估计依赖于去偏的统计场景也可能具有重要价值。