Lifetime data collected from reliability tests are among data that often exhibit significant heterogeneity caused by variations in manufacturing, which makes standard lifetime models inadequate. Finite mixture models provide more flexibility for modeling such data. In this paper, the Weibull‐log‐logistic mixture distributions model is introduced as a new class of flexible models for heterogeneous lifetime data. Some statistical properties of the model are presented including the failure rate function, moments generating function, and characteristic function. The identifiability property of the class of all finite mixtures of Weibull‐log‐logistic distributions is proved. The maximum likelihood estimation (MLE) of model parameters under the Type I and Type II censoring schemes is derived. Some numerical illustrations are performed to study the behavior of the obtained estimators. The model is applied to the hard drive failure data made by the Backblaze data center, where it is found that the proposed model provides more flexibility than the univariate life distributions (Weibull, Exponential, logistic, log‐logistic, Frechet). The failure rate of hard disk drives (HDDs) is obtained based on MLE estimates. The analysis of the failure rate function on the basis of SMART attributes shows that the failure of HDDs can have different causes and mechanisms.

中文翻译：

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威布尔对数-物流混合分布：模型，理论和对生命周期数据的应用

从可靠性测试中收集到的寿命数据属于经常显示由于制造差异而导致明显异质性的数据，这使得标准寿命模型不足。有限的混合模型为建模此类数据提供了更大的灵活性。本文将Weibull-log-logistic混合分布模型作为一类新的用于异质寿命数据的灵活模型引入。提出了模型的一些统计属性，包括失效率函数，矩产生函数和特征函数。证明了Weibull-log-logistic分布的所有有限混合类的可识别性。推导了类型I和类型II审查方案下的模型参数的最大似然估计（MLE）。执行一些数字插图来研究获得的估计量的行为。该模型应用于Backblaze数据中心提供的硬盘驱动器故障数据，发现该模型比单变量寿命分布（Weibull，指数，logistic，log-logistic，Frechet）具有更大的灵活性。硬盘驱动器（HDD）的故障率是根据MLE估算得出的。基于SMART属性的故障率函数分析表明，HDD的故障可能具有不同的原因和机制。硬盘驱动器（HDD）的故障率是根据MLE估算得出的。基于SMART属性的故障率函数分析表明，HDD的故障可能具有不同的原因和机制。硬盘驱动器（HDD）的故障率是根据MLE估算得出的。基于SMART属性的故障率函数分析表明，HDD的故障可能具有不同的原因和机制。