Consistency of semiparametric maximum likelihood estimators for two-phase sampling

Semiparametric maximum likelihood estimators have recently been proposed for a class of two-phase, outcome-dependent sampling models. All of them were "restricted" maximum likelihood estimators, in the sense that the maximization is carried out only over distributions concentrated on the observed values of the covariate vectors. In this paper, the authors give conditions for consistency of these restricted maximum likelihood estimators. They also consider the corresponding unrestricted maximization problems, in which the "absolute" maximum likelihood estimators may then have support on additional points in the covariate space. Their main consistency result also covers these unrestricted maximum likelihood estimators, when they exist for all sample sizes.