On Censoring Time in Statistical Monitoring of Lifetime Data

Life tests for highly reliable products often take a long time even using accelerated life testing with censoring. When the production process is monitored by control charts with the product lifetime as the key quality characteristic, the time spent on life testing could incur significant delays for practitioners to make decisions after sampling. However, shortening the test duration, that results in excessive right-censored observations, inevitably degrades the test power for anomaly detection. This study pays close attention to the determination of censoring time in life tests when monitoring lifetime data with the likelihood-based control charts. To interpret the optimal censoring time, the performance metric—out-of-control average time to signal (OC ATS), is deconstructed into two parts: the original OC ATS and the delay caused by life testing. Finite-sample analytical and large-sample asymptotic expressions of ATS metrics are derived for Type-I censored exponential lifetimes. Similar analytical expressions are also derived for the Weibull case. For general distributions, a Monte Carlo simulation procedure is developed for obtaining approximate results. Our numerical investigation uncovers the 2-fold impact of censoring time on the actual performance of control charts under various scenarios and provides useful references for practitioners to set more sensible censoring times in life testing.

对于高可靠性产品，即便采用带截尾的加速寿命试验（accelerated life testing），寿命试验往往仍耗时良久。当以产品寿命作为关键质量特性，采用控制图（control charts）监控生产过程时，寿命试验所耗费的时长，会导致从业者在抽样后进行决策时面临显著延误。然而，缩短试验时长会导致过多右截尾观测值，不可避免地削弱异常检测的检验效能。本研究聚焦于采用基于似然的控制图（likelihood-based control charts）监控寿命数据时，寿命试验截尾时间的确定问题。为阐释最优截尾时间，本研究将性能指标——失控平均信号时间（out-of-control average time to signal, OC ATS）拆解为两部分：原始失控平均信号时间与寿命试验引发的延迟。针对I型截尾下的指数寿命分布，本研究推导了ATS指标的有限样本解析表达式与大样本渐近表达式。针对威布尔分布（Weibull distribution）场景，本研究亦推导了同类解析表达式。对于一般分布情形，本研究开发了蒙特卡洛模拟（Monte Carlo simulation）流程以获取近似结果。本研究的数值分析揭示了不同场景下截尾时间对控制图实际性能的双重影响，并为从业者在寿命试验中设置更合理的截尾时间提供了有益参考。