Simultaneous confidence bands for survival functions from twice censorship

When a failure time observation is right censored, it is not observed beyond a random right threshold. When a possibly right censored observation is further susceptible to a random left threshold, a twice censored observation results. Patilea and Rolin have proposed product limit estimators for survival functions from twice censored data. Simultaneous confidence bands (SCBs) for survival functions from twice censored data are constructed in a way that mimics the approach that Hollander, McKeague and Yang pursued for random right censoring. Their nonparametric likelihood ratio function is adjusted, providing the basis for constructing the SCBs. The critical value needed for the SCBs is obtained using the bootstrap, for which asymptotic justification is provided. The SCBs align nicely as “neighborhoods” of the Patilea–Rolin nonparametric survival function estimator, in much the same way the likelihood ratio SCBs under random censoring are the “neighborhoods” of the Kaplan–Meier estimator. A simulation study supports the effectiveness of the proposed method. An illustration is given using synthetic data.