Estimation under nearly-correct models

When additional variables are measured on a subsample from an existing cohort, we can fit models using survey-sampling approaches.  In some settings we can also estimate by semiparametric maximum likelihood, profile likelihood, or similar techniques, leading to a semi parametric efficient estimator. For example, under case-control sampling we can use weighted logistic regression or unweighted logistic regression.  The common wisdom is that the weighted estimators are inefficient because of the variation in the weights. I will argue that this is neither true nor helpful. By considering contiguous model misspecification I will show that the efficient estimator gains its extra precision from relying more heavily on the model, and this is true in a quantitative sense, not merely as a heuristic.