  Case-cohort design, an outcome-dependent sampling design for censored
survival data, is increasingly used in biomedical research. The development of
asymptotic theory for a case-cohort design in the current literature primarily
relies on counting process stochastic integrals. Such an approach, however, is
rather limited and lacks theoretical justification for outcome-dependent
weighted methods due to non-predictability. Instead of stochastic integrals, we
derive asymptotic properties for case-cohort studies based on a general
Z-estimation theory for semiparametric models with bundled parameters using
modern empirical processes. Both the Cox model and the additive hazards model
with time-dependent covariates are considered.
