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Models for Imperfect Repair

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Applied Nonparametric Statistics in Reliability

Part of the book series: Springer Series in Reliability Engineering ((RELIABILITY))

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Abstract

We consider two main approaches for modeling of imperfect repair processes. The first approach is via the modeling of conditional intensities, which combines in some sense the features of the “extreme” repair processes, RP and NHPP. The second approach is via the notion of effective ages, where the idea is that repairs may result in improved system behavior equivalent to a certain reduction in time since the system was new. In particular we consider, as part of the first approach, the trend-renewal process (TRP) that is defined in terms of a trend function λ(·) and a renewal distribution F. Nonparametric estimation of λ(·) is considered in the case where F is given on parametric form. For the second approach based on effective ages, we consider in particular nonparametric estimation approaches by Whitaker and Samaniego (J Am Stat Assoc 84:301–309, 1989) and by Peña et al. (J Stat Plan Inference 137:1727–1747, 2007).

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References

  1. Aalen OO, Borgan O, Gjessing HK (2008) Survival and event history analysis. A process point of view. Springer, New York

    Book  MATH  Google Scholar 

  2. Andersen PK, Borgan O, Gill RD, Keiding N (1993) Statistical models based on counting processes. Springer, New York

    MATH  Google Scholar 

  3. Barlow RE, Davis B (1977) Analysis of time between failures of repairable components. In: Fussel JB, Burdick GR (eds) Nuclear systems reliability engineering and risk assessment. SIAM, Philadelphia, pp 543–561

    Google Scholar 

  4. Bartozyński R, Brown BW, McBride CM, Thompson JR (1981) Some nonparametric techniques for estimating the intensity function of a cancer related nonstationary Poisson process. Ann Stat 9(5):1050–1060

    Article  MathSciNet  Google Scholar 

  5. Bathe F, Franz J (1996) Modelling of repairable systems with various degrees of repair. Metrika 43:149–164

    Article  MATH  MathSciNet  Google Scholar 

  6. Berman M (1981) Inhomogeneous and modulated gamma processes. Biometrika 68:143–152

    Article  MATH  MathSciNet  Google Scholar 

  7. Block H, Borges W, Savits T (1985) Age dependent minimal repair. J Appl Probab 22:370–385

    Article  MATH  MathSciNet  Google Scholar 

  8. Brown M, Proschan F (1983) Imperfect repair. J Appl Probab 20:851–859

    Article  MathSciNet  Google Scholar 

  9. Cox DR (1972) The statistical analysis of dependencies in point processes. In: Lewis PA (ed) Stochastic point processes. Wiley, New York, pp 55–66

    Google Scholar 

  10. Dorado C, Hollander M, Sethuraman J (1997) Nonparametric estimation for a general repair model. Ann Stat 25:1140–1160

    Article  MATH  MathSciNet  Google Scholar 

  11. Doyen L, Gaudoin O (2004) Classes of imperfect repair models based on reduction of failure intensity or virtual age. Reliab Eng Syst Saf 84:45–56

    Article  Google Scholar 

  12. Efron B, Tibshirani R (1993) An introduction to the bootstrap. Chapman and Hall/CRC, Boca Raton

    MATH  Google Scholar 

  13. Erlingsen SE (1989) Using reliability data for optimizing maintenance. Unpublished Masters thesis, Norwegian Institute of Technology, Trondheim, Norway

    Google Scholar 

  14. González JR, Slate EH, Peña EA (2009) Package ‘gcmrec’. General class of models for recurrent event data. R package version 1.0-3. http://cran.r-project.org/web/packages/gcmrec/

  15. Heggland K, Lindqvist BH (2007) A non-parametric monotone maximum likelihood estimator of time trend for repairable systems data. Reliab Eng Syst Saf 92:575–584

    Article  Google Scholar 

  16. Hollander M, Presnell B, Sethuraman J (1992) Nonparametric methods for imperfect repair models. Ann Stat 20:879–896

    Article  MATH  MathSciNet  Google Scholar 

  17. Hollander M, Samaniego FJ, Sethuraman J (2007) Imperfect repair. In: Ruggeri F, Kenett R, Faltin FW (eds) Encyclopedia of statistics in quality and reliability. Wiley, New York

    Google Scholar 

  18. Jones MC, Henderson DA (2005) Maximum likelihood kernel density estimation. Technical report 01/05. Department of Statistics, The Open University, United Kingdom

    Google Scholar 

  19. Jones MC, Henderson DA (2009) Maximum likelihood kernel density estimation: on the potential of convolution sieves. Comput Stat Data Anal 53:3726–3733

    Article  MATH  Google Scholar 

  20. Kiefer J, Wolfowitz J (1956) Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters. Ann Math Stat 27:887–906

    Article  MATH  MathSciNet  Google Scholar 

  21. Kijima M (1989) Some results for repairable systems with general repair. J Appl Probab 26:89–102

    Article  MATH  MathSciNet  Google Scholar 

  22. Lakey MJ, Rigdon SE (1992) The modulated power law process. In: Proceedings of the 45th Annual Quality Congress, pp 559–563

    Google Scholar 

  23. Last G, Szekli R (1998) Stochastic comparison of repairable systems by coupling. J Appl Probab 35:348–370

    Article  MATH  MathSciNet  Google Scholar 

  24. Lindqvist B (1999) Repairable systems with general repair. In: Schueller G, Kafka P (eds) Safety and reliability. Proceedings of the European conference on safety and reliability, ESREL ’99, Munich, 13–17 Sep 1999. Balkema, Boston, pp 43–48

    Google Scholar 

  25. Lindqvist BH (2006) On the statistical modelling and analysis of repairable systems. Stat Sci 21:532–551

    Article  MATH  MathSciNet  Google Scholar 

  26. Lindqvist BH (2010) Nonparametric estimation of time trend for repairable systems data. In: Balakrishnan N, Nikulin M, Rykov V (eds) Mathematical and statistical methods in reliability. Applications to medicine, finance, and quality control. Birkhauser, Boston, pp 277–288

    Google Scholar 

  27. Lindqvist BH, Elvebakk G, Heggland K (2003) The trend-renewal process for statistical analysis of repairable systems. Technometrics 45:31–44

    Article  MathSciNet  Google Scholar 

  28. Peña EA (2006) Dynamic modeling and statistical analysis of event times. Stat Sci 21:487–500

    Article  MATH  MathSciNet  Google Scholar 

  29. Peña E, Hollander M (2004) Models for recurrent events in reliability and survival analysis. In: Soyer R, Mazzuchi T, Singpurwalla N (eds) Mathematical reliability: an expository perspective. Kluwer, Boston, pp 105–123

    Google Scholar 

  30. Peña EA, Slate EH, González JR (2007) Semiparametric inference for a general class of models for recurrent events. J Stat Plan Inference 137:1727–1747

    Article  MATH  MathSciNet  Google Scholar 

  31. Ramlau-Hansen H (1983) Smoothing counting process intensities by means of kernel functions. Ann Stat 11:453–466

    Article  MATH  MathSciNet  Google Scholar 

  32. Robertson T, Wright FT, Dykstra RL (1988) Order restricted statistical inference. Wiley, New York

    MATH  Google Scholar 

  33. Smith WL (1958) Renewal theory and its ramifications. J R Stat Soc Ser B 20:243–302

    Google Scholar 

  34. Whitaker LR, Samaniego FJ (1989) Estimating the reliability of systems subject to imperfect repair. J Am Stat Assoc 84:301–309

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Facultad Ciencias, Depto. Estadistica e Investigacion Operativa, Universidad Granada, Campus Univ. Fuentenueva, 18071, Granada, Spain

    Assoc. Prof. M. Luz Gámiz

  2. Department of Mathematical Sciences, Clemson University, Clemson, SC, 29634-1907, USA

    Prof. K. B. Kulasekera

  3. Centre de Recherches de Royallieu, Laboratoire de Mathématiques Appliquées, Université de Technologie de Compiègne, BP 20529 , 60205, Compiègne, France

    Prof. Nikolaos Limnios

  4. Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491, Trondheim, Norway

    Prof. Bo Henry Lindqvist

Authors
  1. Assoc. Prof. M. Luz Gámiz
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  2. Prof. K. B. Kulasekera
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  3. Prof. Nikolaos Limnios
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  4. Prof. Bo Henry Lindqvist
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Correspondence to M. Luz Gámiz .

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© 2011 Springer-Verlag London Limited

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Gámiz, M.L., Kulasekera, K.B., Limnios, N., Lindqvist, B.H. (2011). Models for Imperfect Repair. In: Applied Nonparametric Statistics in Reliability. Springer Series in Reliability Engineering. Springer, London. https://doi.org/10.1007/978-0-85729-118-9_4

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  • DOI: https://doi.org/10.1007/978-0-85729-118-9_4

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  • Print ISBN: 978-0-85729-117-2

  • Online ISBN: 978-0-85729-118-9

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