  In the presence of a missing response, reweighting the complete case
subsample by the inverse of nonmissing probability is both intuitive and easy
to implement. When the population totals of some auxiliary variables are known
and when the inclusion probabilities are known by design, survey statisticians
have developed calibration methods for improving efficiencies of the inverse
probability weighting estimators and the methods can be applied to missing data
analysis. Model-based calibration has been proposed in the survey sampling
literature, where multidimensional auxiliary variables are first summarized
into a predictor function from a working regression model. Usually, one working
model is being proposed for each parameter of interest and results in different
sets of calibration weights for estimating different parameters. This paper
considers calibration using multiple working regression models for estimating a
single or multiple parameters. Contrary to a common belief that overfitting
hurts efficiency, we present three rather unexpected results. First, when the
missing probability is correctly specified and multiple working regression
models for the conditional mean are posited, calibration enjoys an oracle
property: the same semiparametric efficiency bound is attained as if the true
outcome model is known in advance. Second, when the missing data mechanism is
misspecified, calibration can still be a consistent estimator when any one of
the outcome regression models is correctly specified. Third, a common set of
calibration weights can be used to improve efficiency in estimating multiple
parameters of interest and can simultaneously attain semiparametric efficiency
bounds for all parameters of interest. We provide connections of a wide class
of calibration estimators, constructed based on generalized empirical
likelihood, to many existing estimators in biostatistics, econometrics and
survey sampling and perform simulation studies to show that the finite sample
properties of calibration estimators conform well with the theoretical results
being studied.
