We consider asymptotic properties of the nonparametric maximum likelihood estimate (NPMLE) of a failure time distribution function based on doubly interval censored data.In many disease progression or cpidemiological studies, the failure time of interest is the elapsed time between an initial event and a subsequent event and the observations on both events are interval-censored.As a consequence, the es timation is much more complicated than that for right-censored or interval-censored failure time data both theoretically and practically.Although several algorithms have been developed, the asymptotic properties of the estimates are still largely un known.This paper investigates the rate of convergence of the NPMLE and shows that it is slower than that with right-censored or interval-censored data.Further more, we show the asymptotic normality of estimated smooth functionals of the NPMLE, which is useful for the inference of doubly interval-censored data.We also show some asymptotic results on NPMLE based on the multivariate doubly interval censored failure time data.