Reconstruct Kaplan--Meier Estimator as M-estimator and Its Confidence Band

TL;DR

This paper reinterprets the Kaplan--Meier estimator as an M-estimator using an EM algorithm, providing new insights into its properties and enabling the derivation of confidence bands.

Contribution

It introduces a novel M-estimation framework for the KM estimator, linking it with EM algorithm and enabling confidence band construction.

Findings

The EM algorithm converges to the traditional KM estimator.

The M-estimator is statistically equivalent to the KM estimator.

Confidence intervals and bands can be derived from the M-estimator.

Abstract

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.