  We consider a class of doubly weighted rank-based estimating methods for the
transformation (or accelerated failure time) model with missing data as arise,
for example, in case-cohort studies. The weights considered may not be
predictable as required in a martingale stochastic process formulation. We
treat the general problem as a semiparametric estimating equation problem and
provide proofs of asymptotic properties for the weighted estimators, with
either true weights or estimated weights, by using empirical process theory
where martingale theory may fail. Simulations show that the outcome-dependent
weighted method works well for finite samples in case-cohort studies and
improves efficiency compared to methods based on predictable weights. Further,
it is seen that the method is even more efficient when estimated weights are
used, as is commonly the case in the missing data literature. The Gehan
censored data Wilcoxon weights are found to be surprisingly efficient in a wide
class of problems.
