  When estimating causal effects, it is important to assess external validity,
i.e., determine how useful a given study is to inform a practical question for
a specific target population. One challenge is that the covariate distribution
in the population underlying a study may be different from that in the target
population. If some covariates are effect modifiers, the average treatment
effect (ATE) may not generalize to the target population. To tackle this
problem, we propose new methods to generalize or transport the ATE from a
source population to a target population, in the case where the source and
target populations have different sets of covariates. When the ATE in the
target population is identified, we propose new doubly robust estimators and
establish their rates of convergence and limiting distributions. Under
regularity conditions, the doubly robust estimators provably achieve the
efficiency bound and are locally asymptotic minimax optimal. A sensitivity
analysis is provided when the identification assumptions fail. Simulation
studies show the advantages of the proposed doubly robust estimator over simple
plug-in estimators. Importantly, we also provide minimax lower bounds and
higher-order estimators of the target functionals. The proposed methods are
applied in transporting causal effects of dietary intake on adverse pregnancy
outcomes from an observational study to the whole U.S. female population.
