Flexible methods for reliability estimation using aggregate failure-time data

Samira Karimi, Haitao Liao, Neng Fan

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


The actual failure times of individual components are usually unavailable in many applications. Instead, only aggregate failure-time data are collected by actual users, due to technical and/or economic reasons. When dealing with such data for reliability estimation, practitioners often face the challenges of selecting the underlying failure-time distributions and the corresponding statistical inference methods. So far, only the exponential, normal, gamma and inverse Gaussian distributions have been used in analyzing aggregate failure-time data, due to these distributions having closed-form expressions for such data. However, the limited choices of probability distributions cannot satisfy extensive needs in a variety of engineering applications. PHase-type (PH) distributions are robust and flexible in modeling failure-time data, as they can mimic a large collection of probability distributions of non-negative random variables arbitrarily closely by adjusting the model structures. In this article, PH distributions are utilized, for the first time, in reliability estimation based on aggregate failure-time data. A Maximum Likelihood Estimation (MLE) method and a Bayesian alternative are developed. For the MLE method, an Expectation-Maximization algorithm is developed for parameter estimation, and the corresponding Fisher information is used to construct the confidence intervals for the quantities of interest. For the Bayesian method, a procedure for performing point and interval estimation is also introduced. Numerical examples show that the proposed PH-based reliability estimation methods are quite flexible and alleviate the burden of selecting a probability distribution when the underlying failure-time distribution is general or even unknown.

Original languageEnglish (US)
Pages (from-to)101-115
Number of pages15
JournalIISE Transactions
Issue number1
StatePublished - Jan 2 2021


  • Aggregate failure-time data
  • Bayesian method
  • maximum likelihood estimation
  • phase-type distributions

ASJC Scopus subject areas

  • Industrial and Manufacturing Engineering


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