  This paper introduces an assumption-lean method that constructs valid and
efficient lower predictive bounds (LPBs) for survival times with censored data.
We build on recent work by Cand\`es et al. (2021), whose approach first subsets
the data to discard any data points with early censoring times, and then uses a
reweighting technique (namely, weighted conformal inference (Tibshirani et al.,
2019)) to correct for the distribution shift introduced by this subsetting
procedure.
  For our new method, instead of constraining to a fixed threshold for the
censoring time when subsetting the data, we allow for a covariate-dependent and
data-adaptive subsetting step, which is better able to capture the
heterogeneity of the censoring mechanism. As a result, our method can lead to
LPBs that are less conservative and give more accurate information. We show
that in the Type I right-censoring setting, if either of the censoring
mechanism or the conditional quantile of survival time is well estimated, our
proposed procedure achieves nearly exact marginal coverage, where in the latter
case we additionally have approximate conditional coverage. We evaluate the
validity and efficiency of our proposed algorithm in numerical experiments,
illustrating its advantage when compared with other competing methods. Finally,
our method is applied to a real dataset to generate LPBs for users' active
times on a mobile app.
