Abstract:
We demonstrate the semiparametric efficiency of the profile likelihood estimator for population-based case-control studies. We consider in detail four particular designs: the two-phase population-based case-control design (Scott and Wild, Biometrika, 1997), the multi-phase population-based case-control design (Lee, Scott and Wild, Biometrika, 2010), the population-augmented case-control design (Hsieh, Manski and McFadden, Journal of the American Statistical Association, 1985), and the two-stage case-control design (Breslow and Cain, Biometrika, 1988). The efficiency of the estimator for the first three of these designs is explicitly shown and the approach taken can be applied to a general class of designs of which these are members. The results in this thesis are shown to supersede earlier published claims to have demonstrated the semiparametric efficiency of these estimators and, as a consequence, the results in this thesis provide the first complete proof of their semiparametric efficiency. The approach taken is based on extending a nonparametric convolution theorem to cover the case of “multi-sample” models, and then establishing the efficiency of each estimator by demonstrating that the influence function of the estimator is an element of the tangent space.
