  In many important applications of precision medicine, the outcome of interest
is time to an event (e.g., death, relapse of disease) and the primary goal is
to identify the optimal individualized decision rule (IDR) to prolong survival
time. Existing work in this area have been mostly focused on estimating the
optimal IDR to maximize the We propose a new robust framework for estimating an
optimal static or dynamic IDR with time-to-event outcomes based on an
easy-to-interpret quantile criterion. The new method does not need to specify
an outcome regression model and is robust for heavy-tailed distribution. The
estimation problem corresponds to a nonregular M-estimation problem with both
finite and infinite-dimensional nuisance parameters. Employing advanced
empirical process techniques, we establish the statistical theory of the
estimated parameter indexing the optimal IDR. Furthermore, we prove a novel
result that the proposed approach can consistently estimate the optimal value
function under mild conditions even when the optimal IDR is non-unique, which
happens in the challenging setting of exceptional laws. We also propose a
smoothed resampling procedure for inference. The proposed methods are
implemented in the R-package QTOCen. We demonstrate the performance of the
proposed new methods via extensive Monte Carlo studies and a real data
application.restricted mean survival time in the population.
