<p>Left-truncated and interval-censored data often occur in epidemiology and the nonparametric maximum likelihood estimator (NPMLE) of a survival function based on this type of data has been studied in the literature. In this article, we revisit the analysis of left-truncated and interval-censored data based on two sampling methods, leading to two different models. The first model assumes that left-censoring variable is always larger than left-truncation variable while the second model allows left-censoring variable to be smaller than left-truncation variable. Based on the innermost intervals derived by [<CitationRef CitationID="CR38">38</CitationRef>], we obtain the NPMLE of the survival function using self-consistent algorithms. The simulation results demonstrate consistency of the NPMLE under both sampling methods. In addition, under the first sampling method, we consider estimation of the joint distribution function of the duration times between successive events, where the first event is subject to left truncation and interval censoring and the second event is subject to dependent censoring. Based on the NPMLE, we obtain the imputed left-truncated and right-censored data for the first event and the imputed dependent censored data for the second event. Using the imputed data and the inverse-probability-weighted approach, we propose a nonparametric estimator of the joint distribution function of two duration times. The simulation results show that the proposed estimator performs well.</p>