On the solutions to Efron's self-consistency equation
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The main purpose of this note is to provide a new and simple proof that the Kaplan-Meier (i.e., product-limit) estimator for the survival distribution uniquely solves Efron's self-consistency equation. Two novel insights include the observations that (i) the self-consistency equation actually directly generates a Volterra integral equation with solution given by the product-limit estimator for the censoring distribution; and, (ii) the corresponding version of this same integral equation for Kaplan-Meier estimator leads to a simple proof that it can be represented as an inverse probability of censoring weighted estimator.
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