Optimal Adjustment Sets for Nonparametric Estimation of Weighted Controlled Direct Effect

The weighted controlled direct effect (WCDE) generalizes the standard controlled direct effect (CDE) by averaging over the mediator distribution, providing a robust estimate when treatment effects vary across mediator levels. This makes the WCDE especially relevant in fairness analysis, where it isolates the direct effect of an exposure on an outcome, independent of mediating pathways. In this work, we first establish necessary and sufficient conditions for the unique identifiability of the WCDE, clarifying when it diverges from the CDE. Next, we derive the efficient influence function for the WCDE and consider the class of regular and asymptotically linear estimators. We characterize the optimal covariate adjustment set that minimizes asymptotic variance, demonstrating how mediator-confounder interactions introduce distinct requirements compared to average treatment effect estimation. Our results offer a principled framework for efficient estimation of direct effects in complex causal systems, with practical applications in fairness and mediation analysis.