  Multiple imputation provides us with efficient estimators in model-based
methods for handling missing data under the true model. It is also
well-understood that design-based estimators are robust methods that do not
require accurately modeling the missing data; however, they can be inefficient.
In any applied setting, it is difficult to know whether a missing data model
may be good enough to win the bias-efficiency trade-off. Raking of weights is
one approach that relies on constructing an auxiliary variable from data
observed on the full cohort, which is then used to adjust the weights for the
usual Horvitz-Thompson estimator. Computing the optimally efficient raking
estimator requires evaluating the expectation of the efficient score given the
full cohort data, which is generally infeasible. We demonstrate multiple
imputation (MI) as a practical method to compute a raking estimator that will
be optimal. We compare this estimator to common parametric and semi-parametric
estimators, including standard multiple imputation. We show that while
estimators, such as the semi-parametric maximum likelihood and MI estimator,
obtain optimal performance under the true model, the proposed raking estimator
utilizing MI maintains a better robustness-efficiency trade-off even under mild
model misspecification. We also show that the standard raking estimator,
without MI, is often competitive with the optimal raking estimator. We
demonstrate these properties through several numerical examples and provide a
theoretical discussion of conditions for asymptotically superior relative
efficiency of the proposed raking estimator.
