  In prevalent cohort studies with follow-up, the time-to-event outcome is
subject to left truncation leading to selection bias. For estimation of the
distribution of time-to-event, conventional methods adjusting for left
truncation tend to rely on the (quasi-)independence assumption that the
truncation time and the event time are "independent" on the observed region.
This assumption is violated when there is dependence between the truncation
time and the event time possibly induced by measured covariates. Inverse
probability of truncation weighting leveraging covariate information can be
used in this case, but it is sensitive to misspecification of the truncation
model. In this work, we apply the semiparametric theory to find the efficient
influence curve of an expected (arbitrarily transformed) survival time in the
presence of covariate-induced dependent left truncation. We then use it to
construct estimators that are shown to enjoy double-robustness properties. Our
work represents the first attempt to construct doubly robust estimators in the
presence of left truncation, which does not fall under the established
framework of coarsened data where doubly robust approaches are developed. We
provide technical conditions for the asymptotic properties that appear to not
have been carefully examined in the literature for time-to-event data, and
study the estimators via extensive simulation. We apply the estimators to two
data sets from practice, with different right-censoring patterns.
