The Kaplan-Meier Estimator as an Inverse-Probability-of-Censoring Weighted Average

The Kaplan-Meier (product-limit) estimator of the survival function of randomly-censored time-to-event data is a central quantity in survival analysis. It is usually introduced as a nonparametric maximum likelihood estimator, or else as the output of an imputation scheme for censored observations such as redistribute-to-the-right or self-consistency. Following recent work by Robins and Rotnitzky, we show that the Kaplan-Meier estimator can also be represented as a weighted average of identically distributed terms, where the weights are related to the survival function of censoring times. We give two demonstrations of this representation; the first assumes a Kaplan-Meier form for the censoring time survival function, the second estimates the survival functions of failure and censoring times simultaneously and can be developed without prior introduction to the Kaplan-Meier estimator.