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We examine the performance of efficient and AIPW estimators under two-phasesampling when the complete-data model is nearly correctly specified, in thesense that the misspecification is not reliably detectable from the data by anypossible diagnostic or test. Asymptotic results for these nearly true modelsare obtained by representing them as sequences of misspecified models that aremutually contiguous with a correctly specified model. We find that for theleast-favourable direction of model misspecification the bias in the efficientestimator induced can be comparable to the extra variability in the AIPWestimator, so that the mean squared error of the efficient estimator is nolonger lower. This can happen when the most-powerful test for the modelmisspecification still has modest power. We verify that the theoretical resultsagree with simulation in three examples: a simple informative-sampling modelfor a Normal mean, logistic regression in the classical case-control design,and linear regression in a two-phase design.

Robustness of semiparametric efficiency in nearly-true models for two-phase samples