In science and engineering, we are often interested in learning about the lifetime characteristics of the system as well as those of the components that made up the system. However, in many cases, the system lifetimes can be observed but not the component lifetimes, and so we may not also have any knowledge on the structure of the system. Statistical procedures for estimating the parameters of the component lifetime distribution and for identifying the system structure based on system-level lifetime data are developed here using expectation–maximization (EM) algorithm. Different implementations of the EM algorithm based on system-level or component-level likelihood functions are proposed. A special case that the system is known to be a coherent system with unknown structure is considered. The methodologies are then illustrated by considering the component lifetimes to follow a two-parameter Weibull distribution. A numerical example and a Monte Carlo simulation study are used to evaluate the performance and related merits of the proposed implementations of the EM algorithm. Lognormally distributed component lifetimes and a real data example are used to illustrate how the methodologies can be applied to other lifetime models in addition to the Weibull model. Finally, some recommendations along with concluding remarks are provided.

中文翻译：

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系统结构未知的基于系统的生命周期数据的期望最大化算法

在科学和工程领域，我们经常对了解系统以及组成系统的组件的寿命特征感兴趣。但是，在许多情况下，只能观察到系统的使用寿命，而不能观察到组件的使用寿命，因此我们可能对系统的结构也没有任何了解。本文使用期望最大化（EM）算法开发了用于估计组件寿命分布的参数和用于基于系统级寿命数据识别系统结构的统计程序。提出了基于系统级或组件级似然函数的EM算法的不同实现。考虑一种特殊情况，即已知该系统是结构未知的相干系统。然后通过考虑遵循两个参数威布尔分布的组件寿命来说明该方法。数值算例和蒙特卡洛模拟研究用于评估EM算法提出的实现的性能和相关优点。使用对数正态分布的组件寿命和一个实际数据示例来说明如何将该方法应用于除Weibull模型之外的其他寿命模型。最后，提供了一些建议以及结论。使用对数正态分布的组件寿命和一个实际数据示例来说明如何将该方法应用于除Weibull模型之外的其他寿命模型。最后，提供了一些建议以及结论。使用对数正态分布的组件寿命和一个实际数据示例来说明如何将该方法应用于除Weibull模型之外的其他寿命模型。最后，提供了一些建议以及结论。