  A solution to control for nonresponse bias consists of multiplying the design
weights of respondents by the inverse of estimated response probabilities to
compensate for the nonrespondents. Maximum likelihood and calibration are two
approaches that can be applied to obtain estimated response probabilities. We
consider a common framework in which these approaches can be compared. We
develop an asymptotic study of the behavior of the resulting estimator when
calibration is applied. A logistic regression model for the response
probabilities is postulated. Missing at random and unclustered data are
supposed. Three main contributions of this work are: 1) we show that the
estimators with the response probabilities estimated via calibration are
asymptotically equivalent to unbiased estimators and that a gain in efficiency
is obtained when estimating the response probabilities via calibration as
compared to the estimator with the true response probabilities, 2) we show that
the estimators with the response probabilities estimated via calibration are
doubly robust to model misspecification and explain why double robustness is
not guaranteed when maximum likelihood is applied, and 3) we discuss and
illustrate problems related to response probabilities estimation, namely
existence of a solution to the estimating equations, problems of convergence,
and extreme weights. We explain and illustrate why the first aforementioned
problem is more likely with calibration than with maximum likelihood
estimation. We present the results of a simulation study in order to illustrate
these elements.
