Consistency of Semiparametric Maximum Likelihood Estimators for Two-Phase, Outcome Dependent Sampling

Semiparametric maximum likelihood estimators have recently been proposed for a class of two-phase, outcome dependent sampling models; e.g. Breslow and Holubkov (1997), Scott and Wild (1998), and Lawless, Wild, and Kalb eisch (1999). The estimators studied by these authors are predicated on the estimates of the underlying covariate distribution being concentrated on the observed covariate values. Here we give conditions for consistency of the (restricted) maximum likelihood estimators proposed by these authors. We also consider the corresponding maximization problems in further detail and show that the unrestricted maximum likelihood estimators may have support on additional points in the covariate space. In the companion paper by Breslow, McNeney, and Wellner (2000a), eÆciency and asymptotic normality of the restricted maximum likelihood estimators are also established.