The Kaplan-Meier estimator of a survival function is well known to be asymptotically efficient when cause of failure is always observed. It has been an open problem, however, to find an efficient estimator when failure indicators are missing at random. Lo showed that nonparametric maximum likelihood estimators are inconsistent, and this has led to several proposals of ad hoc estimators, none of which are efficient. We now introduce a sieved nonparametric maximum likelihood estimator, and show that it is efficient. Our approach is related to the estimation of a bivariate survival function from bivariate right-censored data.
"Efficient estimation from right-censored data when failure indicators are missing at random." Ann. Statist. 26 (1) 164 - 182, February 1998. https://doi.org/10.1214/aos/1030563981