Lifetime Data Analysis

, Volume 11, Issue 4, pp 511–527 | Cite as

Accelerated Degradation Models for Failure Based on Geometric Brownian Motion and Gamma Processes

  • Chanseok Park
  • W. J. Padgett


Based on a generalized cumulative damage approach with a stochastic process describing degradation, new accelerated life test models are presented in which both observed failures and degradation measures can be considered for parametric inference of system lifetime. Incorporating an accelerated test variable, we provide several new accelerated degradation models for failure based on the geometric Brownian motion or gamma process. It is shown that in most cases, our models for failure can be approximated closely by accelerated test versions of Birnbaum–Saunders and inverse Gaussian distributions. Estimation of model parameters and a model selection procedure are discussed, and two illustrative examples using real data for carbon-film resistors and fatigue crack size are presented.


inverse Gaussian (Wald) distribution degradation process accelerated life test geometric Brownian motion process gamma process censoring 


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  1. H. Akaike, “Information theory and an extension of the maximum likelihood principle,” in Proceedings of the Second International Symposium on Information Theory (B. N. Petrov and F. Czáki, eds.). Budapest: Akademiai Kiadó, pp. 267–281, 1973. Reprinted in Breakthroughs in Statistics, vol. 1(S. Kotz and N. L. Johnson, eds.). Berlin: Springer, pp. 610–624, 1993.Google Scholar
  2. Akaike, H. 1974“A new look at the statistical model identification”IEEE Transactions on Automatic Control19716722zbMATHMathSciNetGoogle Scholar
  3. Bagdonavicius, V., Nikulin, M. 2000“Estimation in degradation models with explanatory variables”Lifetime Data Analysis785103MathSciNetGoogle Scholar
  4. Bagdonavicius, V. 2002Nikulin, Accelerated Life Models, Modeling and Statistical AnalysisChapman & Hall /CRCBoca Raton, FLGoogle Scholar
  5. Bhattacharyya, G. K., Fries, A. 1982“Fatigue failure models – Birnbaum–Saunders vs. inverse Gaussian”IEEE Transactions on Reliability31439440Google Scholar
  6. Birnbaum, Z. W., Saunders, S. C. 1969“A new family of life distributions”Journal of Applied Probability6319327MathSciNetGoogle Scholar
  7. Boulanger, M., Escobar, L. A. 1994“Experimental design for a class of accelerated degradation tests”Technometrics36260272Google Scholar
  8. Burnham, K. P., Anderson, D. R. 2002Model Selection and Multi-Model Inference: A Practical Information-Theoretic ApproachSpringerNew YorkGoogle Scholar
  9. Carey, M. B., Koenig, R. H. 1991“Reliability assessment based on accelerated degradation: a case study”IEEE Transactions on Reliability40499506CrossRefGoogle Scholar
  10. Chhikara, R. S., Folks, J. L. 1989The Inverse Gaussian Distribution: Theory, Methodology, and ApplicationsMarcel DekkerNew YorkGoogle Scholar
  11. Cox, D. R., Miller, H. D. 1965The Theory of Stochastic ProcessesJohn Wiley & SonsNew YorkGoogle Scholar
  12. Doksum, K., Normand, S.- L. T. 1995“Gaussian models for degradation processes – Part I: Methods for the analysis of biomarker data”Lifetime Data Analysis1135144CrossRefMathSciNetGoogle Scholar
  13. Durham, S. D., Padgett, W. J. 1997“A cumulative damage model for system failure with application to carbon fibers and composites”Technometrics393444Google Scholar
  14. Hamada, M. 1995“Analysis of experiments for reliability improvement and robust reliability”Balakrishnan, N. eds. Recent Advances in Life Testing and ReliabilityCRC PressBoca Raton, FLGoogle Scholar
  15. Hudak, S. J.,Jr, Saxena, A., Bucci, R. J., Malcolm, R. C. 1978Development of Standard Methods of Testing and Analyzing Fatigue Crack Growth Rate DataTechnical Report AFML-TR-78-40Westinghouse R & D CenterGoogle Scholar
  16. Jacod, J., Shiryaev, A. N. 1987Limit Theorems for Stochastic ProcessesSpringer-VerlagNew YorkGoogle Scholar
  17. Lawless, J., Crowder, M. 2004“Covariates and random effects in a gamma process model with application to degradation and failure”Lifetime Data Analysis10213227CrossRefMathSciNetGoogle Scholar
  18. Lu, C. J., Meeker, W. Q. 1993“Using degradation measures to estimate a time-to-failure distribution”Technometrics35161174MathSciNetGoogle Scholar
  19. J. Lu, “Degradation Processes and Related Reliability Models,” Ph.D. thesis, McGill University, 1995.Google Scholar
  20. Mann, N. R., Schafer, R. E., Singpurwalla, N. D. 1974Methods for Statistical Analysis of Reliability and Life DataJohn Wiley & SonsNew YorkGoogle Scholar
  21. Meeker, W. Q., Escobar, L. A. 1998Statistical Methods for Reliability DataJohn Wiley & SonsNew YorkGoogle Scholar
  22. Meeker, W. Q., Escobar, L. A., Lu, C. J. 1998“Accelerated degradation tests: modeling and analysis”Technometrics408999Google Scholar
  23. Nelson, W. 1990Accelerated Testing: Statistical Models, Test Plans, and Data AnalysesJohn Wiley & SonsNew YorkGoogle Scholar
  24. Onar, A., Padgett, W. J. 2000“Inverse Gaussian accelerated test models based on cumulative damage”Journal of Statistical Computation and Simulation66233247MathSciNetGoogle Scholar
  25. Owen, W. J., Padgett, W. J. 1999“Accelerated test models for system strength based on Birnbaum–Saunders distributions”Lifetime Data Analysis5133147CrossRefMathSciNetGoogle Scholar
  26. Padgett, W. J., Tomlinson, M. A. 2004“Inference from accelerated degradation and failure data based on Gaussian process models”Lifetime Data Analysis10191206CrossRefMathSciNetGoogle Scholar
  27. Park, C., Padgett, W. J. 2005“New cumulative damage models for failure using stochastic processes as initial damage,”IEEE Transactions on Reliability54530540Google Scholar
  28. Pettit, L. I., Young, K. D. S. 1999“Bayesian analysis for inverse Gaussian lifetime data with measures of degradation”Journal of Statistical Computation and Simulation63217234MathSciNetGoogle Scholar
  29. Prabhu, N. U. 1965Stochastic Processes: Basic Theory and Its ApplicationsMacmillan CompanyNew YorkGoogle Scholar
  30. Shiomi, Y., Yanagisawa, T. 1979,“On distribution parameter during accelerated life test for a carbon film resistor”Bulletin of Electrotechnical Laboratory43330345In JapaneseGoogle Scholar
  31. Suzuki, K., Maki, K., Yokogawa, S. 1993“An analysis of degradation data of a carbon film and the properties of the estimators”Matusita, K.Puri, M. L.Hayakawa, T. eds. Proceedings of the Third Pacific Area Statistical ConferenceZeistThe Netherlands501511Google Scholar
  32. Whitmore, G. A. 1995“Estimating degradation by a wiener diffusion process subject to measurement error”Lifetime Data Analysis1307319CrossRefzbMATHGoogle Scholar
  33. Whitmore, G. A., Crowder, M. J., Lawless, J.F. 1998“Failure inference from a marker process based on a bivariate Wiener model”Lifetime Data Analysis4229251CrossRefGoogle Scholar
  34. Whitmore, G. A., Schenkelberg, F. 1997“Modelling accelerated degradation data using wiener diffusion with a scale transformation”Lifetime Data Analysis32745CrossRefGoogle Scholar
  35. Wilk, M. B., Gnanadesikan, R. 1968“Probability plotting methods for the analysis of data”Biometrika55117Google Scholar
  36. Yanagisawa, T. 1997“Estimation of the degradation of amorphous silicon cells”Microelectronics and Reliability37549554Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of Mathematical SciencesClemson UniversityClemsonUSA
  2. 2.Department of StatisticsUniversity of South CarolinaColumbiaUSA

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