  Censored data, where the event time is partially observed, are challenging
for survival probability estimation. In this paper, we introduce a novel
nonparametric fiducial approach to interval-censored data, including
right-censored, current status, case II censored, and mixed case censored data.
The proposed approach leveraging a simple Gibbs sampler has a useful property
of being "one size fits all", i.e., the proposed approach automatically adapts
to all types of non-informative censoring mechanisms. As shown in the extensive
simulations, the proposed fiducial confidence intervals significantly
outperform existing methods in terms of both coverage and length. In addition,
the proposed fiducial point estimator has much smaller estimation errors than
the nonparametric maximum likelihood estimator. Furthermore, we apply the
proposed method to Austrian rubella data and a study of hemophiliacs infected
with the human immunodeficiency virus. The strength of the proposed fiducial
approach is not only estimation and uncertainty quantification but also its
automatic adaptation to a variety of censoring mechanisms.
