Maximum likelihood analysis of masked data in competing risks models with an environmental stress

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It is an important problem to estimate component reliabilities. For a series system due to cost and time constraints associated with failure analysis, all components cannot be investigated and the cause of failure is narrowed to a subset of components in some cases. When such a case occurs, we say that the cause of failure is masked. It is also necessary in some cases to take account of the influence of an environmental stress on all components. In this paper, we consider 2 and 3-component series systems when the component lifelengths are exponentially distributed and an environmental stress follows either a gamma or an inverse Gaussian distribution. We show that the lifelength of the system and the cause of failure are independent of each other. By comparison between the hazard functions in both models, we see that quite short and long lifelengths are more likely to occur in a gamma model than in an inverse Gaussian one. Assuming that the masking probabilities do not depend on which component actually fails, we show that the likelihood function can be factorized into three parts by a reparametrization. For some special cases, some estimators are given in closed-form. We use the computer failure data to see that our model is useful to analyze the real masked data. As compared with the Kaplan-Meier estimator, our models fit this computer data better than no environmental stress model. Further, we determine a suitable model using AIC. We see that the gamma model is fitted to the data better than the inverse Gaussian one. From a limited simulation study for a 3-component series system, we see that the relative errors of some estimators are inversely proportional to the square root of the expected number of systems whose cause of failure is identified.

Original languageEnglish
Pages (from-to)3389-3396
Number of pages8
JournalIEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
Issue number12
Publication statusPublished - 1 Jan 2004


  • Gamma distribution
  • Inverse gaussian distribution
  • Masking probability
  • Reliability analysis

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