Efficient Nonparametric Inference of Causal Mediation Effects with Nonignorable Missing Confounders

We consider causal mediation analysis with confounders subject to nonignorable missingness in a nonparametric framework. Our approach relies on shadow variables that are associated with the missing confounders but independent of the missingness mechanism. The mediation effect of interest is shown to be a weighted average of an iterated conditional expectation, which motivates our Sieve-based Iterative Outward (SIO) estimator. We derive the rate of convergence and asymptotic normality of the SIO estimator, which do not suffer from the ill-posed inverse problem. Essentially, we show that the asymptotic normality is not affected by the slow convergence rate of nonparametric estimators of nuisance functions. Moreover, we demonstrate that our estimator is locally efficient and attains the semiparametric efficiency bound under certain conditions. We accurately depict the efficiency loss attributable to missingness and identify scenarios in which efficiency loss is absent. We also propose a stable and easy-to-implement approach to estimate asymptotic variance and construct confidence intervals for the mediation effects. Finally, we evaluate the finite-sample performance of our proposed approach through simulation studies, and apply it to the CFPS data to show its practical applicability.