  Many epidemiological and clinical studies aim at analyzing a time-to-event
endpoint. A common complication is right censoring. In some cases, it arises
because subjects are still surviving after the study terminates or move out of
the study area, in which case right censoring is typically treated as
independent or non-informative. Such an assumption can be further relaxed to
conditional independent censoring by leveraging possibly time-varying covariate
information, if available, assuming censoring and failure time are independent
among covariate strata. In yet other instances, events may be censored by other
competing events like death and are associated with censoring possibly through
prognoses. Realistically, measured covariates can rarely capture all such
associations with certainty. For such dependent censoring, often covariate
measurements are at best proxies of underlying prognoses. In this paper, we
establish a nonparametric identification framework by formally admitting that
conditional independent censoring may fail in practice and accounting for
covariate measurements as imperfect proxies of underlying association. The
framework suggests adaptive estimators which we give generic assumptions under
which they are consistent, asymptotically normal, and doubly robust. We
illustrate our framework with concrete settings, where we examine the
finite-sample performance of our proposed estimators via a Monte-Carlo
simulation and apply them to the SEER-Medicare dataset.
