  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.
