On nonparametric maximum likelihood estimation with double truncation

Doubly truncated survival data arise if failure times are observed only within certain time intervals. The nonparametric maximum likelihood estimator is widely used to estimate the underlying failure time distribution. Using a directed graph representation of the data suggested by Vardi (1985), a certain graphical condition holds if and only if the nonparametric maximum likelihood estimate exists and is unique. If this condition does not hold, then such an estimate may exist but need not be unique, so another graphical condition is proposed to check whether such an estimate exists. The conditions are simple to check using existing graphical software. Reanalysis of an AIDS incubation time dataset shows that a nonparametric maximum likelihood estimate does not exist for these data.