  Causal mediation analysis concerns the pathways through which a treatment
affects an outcome. While most of the mediation literature focuses on settings
with a single mediator, a flourishing line of research has examined settings
involving multiple mediators, under which path-specific effects (PSEs) are
often of interest. We consider estimation of PSEs when the treatment effect
operates through K(\geq1) causally ordered, possibly multivariate mediators. In
this setting, the PSEs for many causal paths are not nonparametrically
identified, and we focus on a set of PSEs that are identified under Pearl's
nonparametric structural equation model. These PSEs are defined as contrasts
between the expectations of 2^{K+1} potential outcomes and identified via what
we call the generalized mediation functional (GMF). We introduce an array of
regression-imputation, weighting, and "hybrid" estimators, and, in particular,
two K+2-robust and locally semiparametric efficient estimators for the GMF. The
latter estimators are well suited to the use of data-adaptive methods for
estimating their nuisance functions. We establish the rate conditions required
of the nuisance functions for semiparametric efficiency. We also discuss how
our framework applies to several estimands that may be of particular interest
in empirical applications. The proposed estimators are illustrated with a
simulation study and an empirical example.
