Weighted likelihood for semiparametric models and two-phase stratified samples, with application to cox regression

We consider semiparametric models for which solution of Horvitz-Thompson or inverse probability weighted (IPW) likelihood equations with two-phase stratified samples leads to root N consistent and asymptotically Gaussian estimators of both Euclidean and non-parametric parameters. For Bernoulli (independent and identically distributed) sampling, standard theory shows that the Euclidean parameter estimator is asymptotically linear in the IPW influence function. By proving weak convergence of the IPW empirical process, and borrowing results on weighted bootstrap empirical processes, we derive a parallel asymptotic expansion for finite population stratified sampling. Several of our key results have been derived already for Cox regression with stratified case-cohort and more general survey designs. This paper is intended to help interpret this previous work and to pave the way towards a general Horvitz-Thompson approach to semiparametric inference with data from complex probability samples.