  The Kaplan--Meier (KM) estimator, which provides a nonparametric estimate of
a survival function for time-to-event data, has wide application in clinical
studies, engineering, economics and other fields. The theoretical properties of
the KM estimator including its consistency and asymptotic distribution have
been extensively studied. 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, offering a new interpretation of the KM estimator as an M-estimator.
Theoretical properties including the large-sample variance and limiting
distribution of the KM estimator are established using M-estimation theory.
Simulations and application on two real datasets demonstrate that the proposed
M-estimator is exactly equivalent to the KM estimator, while the confidence
interval and band can be derived as well.
