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

We examine the performance of efficient and AIPW estimators under two-phase sampling 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. We find that for the least-favourable direction of model misspecification the bias in the efficient estimator induced can be comparable to the extra variability in the AIPW estimator, so that the mean squared error of the efficient estimator is no longer lower. This can happen when the most-powerful test for the model misspecification still has modest power. We verify that the theoretical results agree with simulation in three examples: a simple informative-sampling model for a Normal mean, logistic regression in the classical case-control design, and linear regression in a two-phase design.