AStA Advances in Statistical Analysis

, Volume 103, Issue 1, pp 69–98 | Cite as

Expectation–maximization algorithm for system-based lifetime data with unknown system structure

  • Yandan Yang
  • Hon Keung Tony NgEmail author
  • Narayanaswamy Balakrishnan
Original Paper


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.


Latent variable Likelihood inference Lognormal distribution Truncated distribution Weibull distribution 



The authors thank two anonymous referees for their careful review and useful comments and suggestions on an earlier version of this manuscript which resulted in this improved version. HKTNs work was supported by a Grant from the Simons Foundation (#280601).


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Statistical ScienceSouthern Methodist UniversityDallasUSA
  2. 2.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

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