Two bias-corrected Kaplan-Meier estimators

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.