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Covariates and Random Effects in a Gamma Process Model with Application to Degradation and Failure

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Abstract

The gamma process is a natural model for degradation processes in which deterioration is supposed to take place gradually over time in a sequence of tiny increments. When units or individuals are observed over time it is often apparent that they degrade at different rates, even though no differences in treatment or environment are present. Thus, in applying gamma-process models to such data, it is necessary to allow for such unexplained differences. In the present paper this is accomplished by constructing a tractable gamma-process model incorporating a random effect. The model is fitted to some data on crack growth and corresponding goodness-of-fit tests are carried out. Prediction calculations for failure times defined in terms of degradation level passages are developed and illustrated.

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References

  • O. Aalen and H. K. Gjessing,#x201C;Understanding the shape of the hazard rate: a process point of view (with discussion)#x201D;, Statistical Science vol. 16 pp. 1–22, 2001.

    Google Scholar 

  • M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover: New York, 1965.

    Google Scholar 

  • P. K. Andersen, O. Borgan, R. D. Gill and N. Keiding, Statistical Models Based on Counting Processes, Springer-Verlag: New York, 2001.

    Google Scholar 

  • V. Bagdonavicius and M. Nikulin,#x201C;Estimation in degradation models with explanatory variables,#x201D; Lifetime Data Analysis vol. 7 pp. 85–103, 2001.

    Google Scholar 

  • E. Cinlar,#x201C;On a generalization of gamma processes,#x201D; Journal of Applied Probability vol. 17 pp. 467–480, 1980.

    Google Scholar 

  • M. J. Crowder and D. J. Hand, Analysis of Repeated Measures, Chapman and Hall: London, 1990.

    Google Scholar 

  • A. C. Davison and D. V. Hinkley, Bootstrap Methods and Their Application, Cambridge University Press, Cambridge: U.K., 1997.

    Google Scholar 

  • K. A. Doksum, and A. Hoyland,#x201C;Models for variable-stress accelerated life test experiments based on Wiener processes and the inverse Gaussian distribution,#x201D; Technometrics vol. 34 pp. 74–82, 1992.

    Google Scholar 

  • R. L. Dykstra and P. Laud,#x201C;A Bayesian nonparametric approach to reliability,#x201D; Annals of Statistics vol. 9 pp. 356–367, 1981.

    Google Scholar 

  • T. S. Ferguson and M. J. Klass,#x201C;A representation of independent increments processes without Gaussian components,#x201D; Annals of Mathematical Statistics vol. 43 pp. 1634–1643, 1972.

    Google Scholar 

  • S. J. Hudak, A. Saxena, R. J. Bucci, and R. C. Malcolm,#x201C;Development of standard methods of testing and analyzing fatigue crack growth rate data,#x201D; Technical Report AFML-TR-78-40, Westinghouse R & D Center, Westinghouse Electric Corporation, Pittsburgh, PA, 1978.

    Google Scholar 

  • J. D. Kalbfleisch,#x201C;Bayesian nonparametric analysis of survival data,#x201D; Journal of the Royal Statistical Society B vol. 40 pp. 214–221, 1978.

    Google Scholar 

  • C. J. Lu and W. Q. Meeker#x201C;Using degradation measurements to estimate a time-to-failure distribution,#x201D; Technometrics vol. 35 pp. 161–174, 1993.

    Google Scholar 

  • W. Q. Meeker and L. E. Escobar, Statistical Methods for Reliability Data, John Wiley and Sons: New York, 1998.

    Google Scholar 

  • P. A. P. Moran, The Theory of Storage, Methuen: London, 1959.

    Google Scholar 

  • M. E. Robinson and M. J. Crowder#x201C;Bayesian methods for a growth-curve degradation model with repeated measures,#x201D; Lifetime Data Analysis vol. 6 pp. 357–374, 2000.

    Google Scholar 

  • N. D. Singpurwalla,#x201C;Survival in dynamic environments,#x201D; Statistical Science vol. 1 pp. 86–103, 1995.

    Google Scholar 

  • D. A. Virkler, B. M. Hilberry, and P. K. Goel#x201C;The statistical nature of fatigue crack propagation,#x201D; Trans. ASME pp. 148–153, 1979.

  • G. A. Whitmore,#x201C;Normal-gamma mixtures of inverse Gaussian distributions,#x201D; Scandanavian Journal of Statistics vol. 13 pp. 211–220, 1986.

    Google Scholar 

  • G. A. Whitmore,#x201C;Estimating degradation by a Wiener diffusion process,#x201D; Lifetime Data Analysis vol. 1 pp. 307–319, 1995.

    Google Scholar 

  • G. A. Whitmore and F. Schenkelberg,#x201C;Modelling accelerated degradation data using Wiener diffusion with a time scale transformation,#x201D; Lifetime Data Analysis vol. 3 pp. 27–45, 1997.

    Google Scholar 

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Authors and Affiliations

  1. Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Jerry Lawless

  2. Department of Mathematics, Imperial College, 180 Queen's Gate, London, SW7 2BZ, UK

    Martin Crowder

Authors
  1. Jerry Lawless
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  2. Martin Crowder
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Lawless, J., Crowder, M. Covariates and Random Effects in a Gamma Process Model with Application to Degradation and Failure. Lifetime Data Anal 10, 213–227 (2004). https://doi.org/10.1023/B:LIDA.0000036389.14073.dd

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  • DOI: https://doi.org/10.1023/B:LIDA.0000036389.14073.dd

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