Large sample theory for semiparametric regression models with two-phase, outcome dependent sampling

Outcome-dependent, two-phase sampling designs can dramatically reduce the costs of observational studies by judicious selection of the most informative subjects for purposes of detailed covariate measurement. Here we derive asymptotic information bounds and the form of the eÆcient score and in uence functions for the semiparametric regression models studied by Lawless, Kalbfleisch, and Wild (1999) under two-phase sampling designs. We relate the eÆcient score to the least-favorable parametric submodel by use of formal calculations suggested by Newey (1994). We then proceed to show that the maximum likelihood estimators proposed by Lawless, Kalbfleisch, and Wild (1999) for both the parametric and nonparametric parts of the model are asymptotically normal and eÆcient, and that the eÆcient in uence function for the parametric part agrees with the more general calculations of Robins, Hsieh, and Newey (1995)