On the Breslow-Holubkov estimator

Breslow and Holubkov (J Roy Stat Soc B 59:447-461 1997a) developed semiparametric maximum likelihood estimation for two-phase studies with a case-control first phase under a logistic regression model and noted that, apart for the overall intercept term, it was the same as the semiparametric estimator for two-phase studies with a prospective first phase developed in Scott and Wild (Biometrica 84:57-71 1997). In this paper we extend the Breslow-Holubkov result to general binary regression models and show that it has a very simple relationship with its prospective first-phase counterpart. We also explore why the design of the first phase only affects the intercept of a logistic model, simplify the calculation of standard errors, establish the semiparametric efficiency of the Breslow-Holubkov estimator and derive its asymptotic distribution in the general case.