The Kaplan-Meier estimator (KME) is a classical nonparametric reliability estimator for incomplete data. Although it has been widely used, its two drawbacks have not been addressed well in the literature: (a) as a staircase function, it actually has two reliability estimates for each failure observation, and (b) it is biased. This paper aims to address these two issues. First, an ideal reliability estimator for complete data is defined and used as a benchmark to quantitatively evaluate the performance of a nonparametric reliability estimator. Then, two bias-corrected KMEs are proposed. One is a weighted average of the two-point moving averaged KME and the modified KME and defined at the failure observations; the other is also a weighted KME but defined at the midpoint of two successive failure observations. Through combining these two estimators, a data-augmented estimator can be obtained, which is particularly useful for parameter estimation on heavily censored data. It is shown that the proposed estimators are almost unbiased and can be conveniently implemented in a spreadsheet program.

中文翻译：

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两个偏差校正的 Kaplan-Meier 估计量

Kaplan-Meier 估计器 (KME) 是用于不完整数据的经典非参数可靠性估计器。尽管它已被广泛使用，但它的两个缺点在文献中并未得到很好的解决：（a）作为一个阶梯函数，它实际上对每个故障观察都有两个可靠性估计，以及（b）它是有偏差的。本文旨在解决这两个问题。首先，定义了完整数据的理想可靠性估计量，并将其用作定量评估非参数可靠性估计量性能的基准。然后，提出了两个经过偏差校正的 KME。一个是两点移动平均 KME 和修改后的 KME 的加权平均值，并在故障观察时定义；另一个也是加权 KME，但定义在两个连续失败观察的中点。通过结合这两个估计器，可以得到一个数据增强的估计器，这对于严重删失数据的参数估计特别有用。结果表明，所提出的估计器几乎是无偏的，并且可以在电子表格程序中方便地实现。