Inferences for case-control and semiparametric two-sample density ratio models

We consider inference in general binary response regression models under retrospective sampling plans. Prentice & Pyke (1979) discovered that inference for the odds-ratio parameter in a logistic model can be based on a prospective likelihood even though the sampling scheme is retrospective. We show that the estimating function obtained from the prospective likelihood is optimal in a class of unbiased estimating functions. Also we link casecontrol sampling with a two-sample biased sampling problem, where the ratio of two densities is assumed to take a known parametric form. Connections between this model and the Cox proportional hazards model are pointed out. Large and small sample size behaviour of the proposed estimators is studied.