Augmented inverse-probability weighted (AIPW) estimators for incomplete-data models typically do not have full semiparametric efficiency, but do have model-robustness properties not shared by the efficient estimator. We examine the performance of efficient and AIPW estimators when the complete-data model is nearly correctly specified, in the sense that the misspecification is not reliably detectable from the data by any possible diagnostic or test. Asymptotic results for these nearly true models are obtained by representing them as sequences of misspecified models that are mutually contiguous with a correctly specified model. For some least favorable direction of model misspecification the bias in the efficient estimator induced by even this amount of model misspecification is comparable to the extra variability in the AIPW estimator, so that the mean squared error of the efficient estimator is no longer lower, at least in a local asymptotic minimax sense.



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