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Reconstructing the Kaplan–Meier Estimator as an M-estimator
The American Statistician ( IF 1.8 ) Pub Date : 2021-07-26 , DOI: 10.1080/00031305.2021.1947376
Jiaqi Gu 1 , Yiwei Fan 1 , Guosheng Yin 1
Affiliation  

Abstract

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.



中文翻译:

将 Kaplan-Meier 估计量重构为 M 估计量

摘要

Kaplan-Meier (KM) 估计器为事件发生时间数据提供生存函数的非参数估计,在临床研究、工程、经济学和许多其他领域具有广泛的应用。KM 估计量的理论性质,包括其一致性和渐近分布,已经得到很好的建立。从一个新的角度来看,我们通过最大化基于一致性的二次 M 函数,将 KM 估计器重构为 M 估计器,这可以使用期望最大化 (EM) 算法来计算。结果表明,EM算法的收敛点与传统的KM估计量一致,为将KM估计量作为M-估计量提供了新的解释。因此,可以使用 M 估计理论建立 KM 估计量的极限分布。

更新日期:2021-07-26
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