  During the past few decades, missing-data problems have been studied
extensively, with a focus on the ignorable missing case, where the missing
probability depends only on observable quantities. By contrast, research into
non-ignorable missing data problems is quite limited. The main difficulty in
solving such problems is that the missing probability and the regression
likelihood function are tangled together in the likelihood presentation, and
the model parameters may not be identifiable even under strong parametric model
assumptions. In this paper we discuss a semiparametric model for non-ignorable
missing data and propose a maximum full semiparametric likelihood estimation
method, which is an efficient combination of the parametric conditional
likelihood and the marginal nonparametric biased sampling likelihood. The extra
marginal likelihood contribution can not only produce efficiency gain but also
identify the underlying model parameters without additional assumptions. We
further show that the proposed estimators for the underlying parameters and the
response mean are semiparametrically efficient. Extensive simulations and a
real data analysis demonstrate the advantage of the proposed method over
competing methods.
