Satten Glen A, Datta Somnath
Division of HIV/AIDS Prevention - Surveillance and Epidemiology, National Center for HIV, STD and TB Prevention, Centers for Disease Control and Prevention, Atlanta, GA 30333 USA.
Department of Statistics, University of Georgia, Athens, GA 30602 USA.
Am Stat. 2001;55(3):207-210. doi: 10.1198/000313001317098185. Epub 2012 Jan 1.
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.
随机删失的事件发生时间数据的生存函数的Kaplan-Meier（乘积限）估计量是生存分析中的核心量。它通常被引入作为非参数最大似然估计量，或者作为对删失观测值的一种插补方案（如右移再分配或自一致性）的输出。继罗宾斯和罗特尼茨基最近的工作之后，我们表明Kaplan-Meier估计量也可以表示为同分布项的加权平均，其中权重与删失时间的生存函数有关。我们给出了这种表示的两种证明；第一种假设删失时间生存函数具有Kaplan-Meier形式，第二种同时估计失效时间和删失时间的生存函数，并且无需事先引入Kaplan-Meier估计量即可展开。
