A model for competing (resp. complementary) risks survival data where the failure time can be left (resp. right) censored is proposed. Product-limit estimators for the survival functions of the individual risks are derived. We deduce the strong convergence of our estimators on the whole real half-line without any additional assumptions and their asymptotic normality under conditions concerning only the observed distribution. When the observations are generated according to the double censoring model introduced by Turnbull, the product-limit estimators represent upper and lower bounds for Turnbull’s estimator.
"Product-limit estimators of the survival function with twice censored data." Ann. Statist. 34 (2) 925 - 938, April 2006. https://doi.org/10.1214/009053606000000065