Reliability estimation from left-truncated and right-censored data using splines

Reliability data are often left truncated and right censored, because the data-collection process usually starts much later than the installation of the first product unit, and some units are still in service at the end of the data collection. The truncation introduces a sampling bias, making analyses of the lifetime data complicated. This study develops a nonparametric likelihood-based estimation procedure for left-truncated and right-censored data using B-splines. In terms of small-sample performance and large-sample efficiencies, the proposed spline-based estimators for the reliability function are shown to be more efficient than the existing nonparametric estimators. We further consider nonparametric two-sample tests for left-truncated and right-censored data. The new class of tests is useful for comparing the reliability of similar products. The test statistics are based on the cumulative weighted differences between the two estimated failure rates. Asymptotic distributions of the proposed statistics are derived and their finite-sample properties are evaluated using Monte Carlo simulations. The performance of the proposed test statistic is compared with that of the weighted Kaplan-Meier statistic. Lastly, a real-life example of high-voltage power transformers is used to illustrate the proposed method.