On the solutions to Efron's self-consistency equation

The purpose of this note is to demonstrate how an important Volterra integral equation leads to a direct proof of the facts that (i) the product-limit estimator for the survival distribution uniquely solves Efron's self-consistency equation; and, (ii) the product-limit estimator for the censoring distribution arises directly as an immediate by-product of the self-consistency equation. As a by-product, we further illustrate how this same Volterra equation results in a simple proof that the Kaplan-Meier estimator can be represented as an inverse probability of censoring weighted estimator.