  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.
