Non‐parametric estimation of gap time survival functions for ordered multivariate failure time data

Times between sequentially ordered events (gap times) are often of interest in biomedical studies. For example, in a cancer study, the gap times from incidence-to-remission and remission-to-recurrence may be examined. Such data are usually subject to right censoring, and within-subject failure times are generally not independent. Statistical challenges in the analysis of the second and subsequent gap times include induced dependent censoring and non-identifiability of the marginal distributions. We propose a non-parametric method for constructing one-sample estimators of conditional gap-time specific survival functions. The estimators are uniformly consistent and, upon standardization, converge weakly to a zero-mean Gaussian process, with a covariance function which can be consistently estimated. Simulation studies reveal that the asymptotic approximations are appropriate for finite samples. Methods for confidence bands are provided. The proposed methods are illustrated on a renal failure data set, where the probabilities of transplant wait-listing and kidney transplantation are of interest.