Abstract: Interval-censored failure time data occur in many fields such as demography, economics, medical research and reliability, and many inference procedures on them have been developed (Chen et al., 2012; Sun, 2006). However, most of the existing approaches assume that the mechanism that yields interval censoring is independent of the failure time of interest and it is clear that this may not be true in practice. In this talk, we will discuss this latter situation and present some inference procedures for the problem.