Doubly-robust estimators are widely used to draw inference about the average effect of a treatment. Such estimators are consistent for the effect of interest if either one of two nuisance parameters is consistently estimated. However, if flexible, data-adaptive estimators of these nuisance parameters are used, double-robustness does not readily extend to inference. We present a general theoretical study of the behavior of doubly-robust estimators of an average treatment effect when one of the nuisance parameters is inconsistently estimated. We contrast different approaches for constructing such estimators and investigate the extent to which they may be modified to also allow doubly-robust inference. We find that while targeted maximum likelihood estimation can be used to solve this problem very naturally, common alternative frameworks appear to be inappropriate for this purpose. We provide a theoretical study and a numerical evaluation of the alternatives considered. Our simulations highlight the need and usefulness of these approaches in practice, while our theoretical developments have broad implications for the construction of estimators that permit doubly-robust inference in other problems.



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