By approximating the classical product-limit estimator of a distribution function with an average of iid random variables, we derive sufficient and necessary conditions for the rate of (both strong and weak) uniform convergence of the product-limit estimator over the whole line. These findings somehow fill a longstanding gap in the asymptotic theory of survival analysis. The result suggests a natural way of estimating the rate of convergence. We also prove a related conjecture raised by Gill and discuss its application to the construction of a confidence interval for a survival function near the endpoint.
"On the rate of uniform convergence of the product-limit estimator: strong and weak laws." Ann. Statist. 25 (3) 1050 - 1087, June 1997. https://doi.org/10.1214/aos/1069362738