Two bias‐corrected Kaplan‐Meier estimators

Abstract

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.

An aging‐intensity‐function‐based parameter estimation method on heavily censored data

“…Two drawbacks with KME are (a) it is biased, and (b) it is a staircase function so that the estimates at failure times are not unique. Jiang 23 proposes two bias‐corrected KMEs to fix these problems. They are weighted KME and mid‐point KME.…”

“…The empirical TTT plot is the plot of $${\varphi _i}$$ versus $${x_i} = {F_i} = 1 - {R_i}$$. When the empirical TTT plot is concave, Jiang 23 uses the following relation to smooth the plot: $$$$\begin{equation}{\varphi _0}\left( x \right) = a{x^{0.5}} + bx + c\ln (1 + x).\end{equation}$$$$…”

“…According to 23 and, 30 for $$\beta = {\beta _e} > 1$$, the FRF of the Weibull distribution is increasing with $$r( \infty ) = \infty $$; the FRF of the gamma distribution is increasing with $$r( \infty ) = 1/v$$; the FRF of the normal distribution is increasing and asymptotically linear; and the FRF of the lognormal distribution is unimodal with $$r( \infty ) = 0$$. Figure 4 illustrates these conclusions; and shows that it is easy for the failure rate curve to differentiate the lognormal and gamma distributions from the Weibull and normal distributions though it is hard to differentiate the Weibull distribution from the normal distribution.…”

“…The empirical TTT plot is the plot of 𝜑 𝑖 versus 𝑥 𝑖 = 𝐹 𝑖 = 1 − 𝑅 𝑖 . When the empirical TTT plot is concave, Jiang 23 uses the following relation to smooth the plot:…”

“…According to 23 and, 30 for 𝛽 = 𝛽 𝑒 > 1, the FRF of the Weibull distribution is increasing with 𝑟(∞) = ∞; the FRF of the gamma distribution is increasing with 𝑟(∞) = 1∕𝑣; the FRF of the normal distribution is increasing and asymptotically linear; and the FRF of the lognormal distribution is unimodal with 𝑟(∞) = 0. Figure 4 illustrates these conclusions; and…”