Reconstructing the Kaplan–Meier Estimator as an M-estimator

The Kaplan–Meier (KM) estimator, which provides a nonparametric estimate of a survival function for time-to-event data, has broad applications in clinical studies, engineering, economics and many other fields. The theoretical properties of the KM estimator including its consistency and asymptotic distribution have been well established. From a new perspective, we reconstruct the KM estimator as an M-estimator by maximizing a quadratic M-function based on concordance, which can be computed using the expectation–maximization (EM) algorithm. It is shown that the convergent point of the EM algorithm coincides with the traditional KM estimator, which offers a new interpretation of the KM estimator as an M-estimator. As a result, the limiting distribution of the KM estimator can be established using M-estimation theory. Application on two real datasets demonstrates that the proposed M-estimator is equivalent to the KM estimator, and the confidence intervals and confidence bands can be derived as well.