  We propose a general index model for survival data, which generalizes many
commonly used semiparametric survival models and belongs to the framework of
dimension reduction. Using a combination of geometric approach in
semiparametrics and martingale treatment in survival data analysis, we devise
estimation procedures that are feasible and do not require
covariate-independent censoring as assumed in many dimension reduction methods
for censored survival data. We establish the root-$n$ consistency and
asymptotic normality of the proposed estimators and derive the most efficient
estimator in this class for the general index model. Numerical experiments are
carried out to demonstrate the empirical performance of the proposed estimators
and an application to an AIDS data further illustrates the usefulness of the
work.
