Nonparametric estimation of distribution functions with left truncated and doubly censored data

A common objective of survival studies is to estimate the cumulative distribution function (CDF) of duration time between two successive events, an initial event and a failure event. Cohort study usually includes the individuals who have experienced the initial event but not the failure event, leading to left truncated and doubly interval-censored (LTDC) data if both events are subject to interval censoring. In this article, we consider the situation when the initial event is subject to left truncation and partly interval censoring and the failure event is subject to left truncation and interval censoring. We propose a three-step procedure for estimating the CDF of the failure time, denoted as F. In the first step, we obtain the nonparametric maximum likelihood estimator (NPMLE) of F based solely on data with the exact calendar times of the initial event. In the second and third steps, we consider two approaches, leading to two different estimators. The first approach is based on maximization of the conditional likelihood function of the initial event given the NPMLE from the first step and that of the failure time given the estimated CDF of the initial event from the second step. The second approach imputes the estimated conditional mean for the failure event in the second step and that for the initial events in the third step. The proposed method is a generalization of the existing method which considers the cases when the initial event is subject to interval censoring and failure time is discrete. Simulation results indicate that the proposed estimators perform well with finite sample sizes.