A self-consistent estimator of survival function with interval-censored and left-truncated data

Interval censoring refers to a situation in which, Ti∗, the time to occurrence of an event of interest is only known to lie in an interval [Li∗,Ri∗]. In some cases, the variable Ti∗ also suffers left-truncation. The nonparametric maximum likelihood estimator (NPMLE) of the survival function of Ti∗ can be obtained by using an EM algorithm of Turnbull (1976). One disadvantage of the NPMLE is that it is not uniquely defined in the innermost intervals. In this article, we propose a self-consistent estimator (SCE), which does not require interpolation. Furthermore, we show that the NPMLE is also an SCE. We establish the consistency of the SCE under certain conditions, which implies that the NPMLE is also a consistent estimator. A simulation study is conducted to compare the performance between the SCE and the NPMLE.