Fitting semi-Markov models to interval-censored data with unknown initiation times

In a semi-Markov model, the hazard of making a transition between stages depends on the time spent in the current stage but is independent of time spent in other stages. If the initiation time (time of entry into the network) is not known for some persons and if transition time data are interval censored (i.e., if transition times are not known exactly but are known only to have occurred in some interval), then the length of time these persons spent in any stage is not known. We show how a semi-Markov model can still be fit to interval-censored data with missing initiation times. For the special case of models in which all persons enter the network at the same initial stage and proceed through the same succession of stages to a unique absorbing stage, we present discrete-time nonparametric maximum likelihood estimators of the waiting-time distributions for this type of data.