• Chain-of-events data;
  • EM algorithm;
  • Interval censoring;
  • Multistage model;
  • Self-consistent estimator;
  • Semi-Markov assumption;
  • Truncation

Summary. Chain-of-events data are longitudinal observations on a succession of events that can only occur in a prescribed order. One goal in an analysis of this type of data is to determine the distribution of times between the successive events. This is difficult when individuals are observed periodically rather than continuously because the event times are then interval censored. Chain-of-events data may also be subject to truncation when individuals can only be observed if a certain event in the chain (e.g., the final event) has occurred. We provide a nonparametric approach to estimate the distributions of times between successive events in discrete time for data such as these under the semi-Markov assumption that the times between events are independent. This method uses a self-consistency algorithm that extends Turnbull's algorithm (1976, Journal of the Royal Statistical Society, Series B38, 290–295). The quantities required to carry out the algorithm can be calculated recursively for improved computational efficiency. Two examples using data from studies involving HIV disease are used to illustrate our methods.