  Quantile regression has been advocated in survival analysis to assess
evolving covariate effects. However, challenges arise when the censoring time
is not always observed and may be covariate-dependent, particularly in the
presence of continuously-distributed covariates. In spite of several recent
advances, existing methods either involve algorithmic complications or impose a
probability grid. The former leads to difficulties in the implementation and
asymptotics, whereas the latter introduces undesirable grid dependence. To
resolve these issues, we develop fundamental and general quantile calculus on
cumulative probability scale in this article, upon recognizing that probability
and time scales do not always have a one-to-one mapping given a survival
distribution. These results give rise to a novel estimation procedure for
censored quantile regression, based on estimating integral equations. A
numerically reliable and efficient Progressive Localized Minimization (PLMIN)
algorithm is proposed for the computation. This procedure reduces exactly to
the Kaplan--Meier method in the $k$-sample problem, and to standard uncensored
quantile regression in the absence of censoring. Under regularity conditions,
the proposed quantile coefficient estimator is uniformly consistent and
converges weakly to a Gaussian process. Simulations show good statistical and
algorithmic performance. The proposal is illustrated in the application to a
clinical study.
