Advertisement

Models for Minimal Repair

  • M. Luz Gámiz
  • K. B. Kulasekera
  • Nikolaos Limnios
  • Bo Henry Lindqvist
Chapter
Part of the Springer Series in Reliability Engineering book series (RELIABILITY)

Abstract

In this chapter, we consider systems that, once the failure has occurred and the repair has been completed, exhibit identical conditions to that just before the failure. In this case, minimal repair maintenance actions are being carried out in the system environment. The model considered is a non-homogeneous Poisson process (NHPP) and the main objective is a nonparametric estimation of the intensity function of the process or equivalently, the rate occurrence of failures (rocof). To achieve this, we will consider one or multiple realizations of a NHPP, which means observing a single system or, alternatively, a population of systems of the same characteristics. No assumption is adopted for the functional form of the rocof except concerning the smoothness, which is understood in terms of some properties of derivability. We are also interested in estimating this function under some restrictions of monotonicity.

Keywords

Failure Time Intensity Function Counting Process Confidence Band Bandwidth Parameter 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Aalen O (1978) Nonparametric inference for a family of counting processes. Ann Stat 6(4):701–726MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Andersen P, Borgan O, Gill R, Keiding N (1993) Statistical models based on counting processes. Springer, New YorkMATHGoogle Scholar
  3. 3.
    Ascher H (1968) Evaluation of a repairable system reliability using ‘bad-as-old’ concept. IEEE Trans Reliab 17:103–110CrossRefGoogle Scholar
  4. 4.
    Ascher H (2007a) Repairable systems reliability. In: Ruggeri F, Kenett R, Faltin FW (eds) Encyclopedia of statistics in quality and reliability. Wiley, New YorkGoogle Scholar
  5. 5.
    Ascher H (2007b) Different insights for improving part and system reliability obtained from exactly same DFOM ‘failure numbers’. Reliab Eng Syst Saf 92:552–559CrossRefGoogle Scholar
  6. 6.
    Ascher H, Feingold H (1984) Repairable systems reliability: modeling, inference, misconceptions and their causes. Marcel Dekker, New YorkMATHGoogle Scholar
  7. 7.
    Aven T (2007) General minimal repair models. In: Ruggeri F, Kenett R, Faltin FW (eds) Encyclopedia of statistics in quality and reliability. Wiley, New YorkGoogle Scholar
  8. 8.
    Bartoszyński R, Brown BW, McBride M, Thompson JR (1981) Some nonparametric techniques for estimating the intensity function of a cancer related nonstationary Poisson process. Ann Stat 9(5):1050–1060MATHCrossRefGoogle Scholar
  9. 9.
    Boswell MT (1966) Estimating and testing trend in a stochastic process of Poisson type. Ann Math Stat 37(6):1564–1573MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Bowman AW (1984) An alternative method of cross-validation for the smoothing of density estimates. Biometrika 71:353–360CrossRefMathSciNetGoogle Scholar
  11. 11.
    Brooks MM, Marron JS (1991) Asymptotic optimality of the least-squares cross-validation bandwidth for kernel estimates of density functions. Stoch Process Appl 38:157–165MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Chiang CT, Wang M-C, Huang CY (2005) Kernel estimation of rate function for recurrent event data. Scand J Stat 32(1):77–91MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Cowling A, Hall P, Phillips MJ (1996) Bootstrap confidence regions for the intensity of a Poisson point process. J Am Stat Assoc 91:1516–1524MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Diggle PJ, Marron JS (1988) Equivalence of smoothing parameter selectors in density and intensity estimation. J Am Stat Assoc 83:793–800MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Finkelstein M (2008) Failure rate modelling for reliability and risk. Springer, LondonGoogle Scholar
  16. 16.
    González JR, Slate EH, Peña EA (2009) gcmrec: general class of models for recurrent event data http://www.r-project.orgGoogle Scholar
  17. 17.
    Henderson SG (2003) Estimation for nonhomogeneous Poisson process from aggregated data. Oper Res Lett 31:375–382MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    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 YorkGoogle Scholar
  19. 19.
    Krivtsov VV (2006) Practical extensions to NHPP application in repairable system reliability analysis. Reliab Eng Syst Saf 92:560–562CrossRefGoogle Scholar
  20. 20.
    Kumar U, Klefsjö B (1992) Reliability analysis of hydraulic systems of LHD machines using the power law process model. Reliab Eng Syst Saf 35:217–224CrossRefGoogle Scholar
  21. 21.
    Leadbetter MR, Wold D (1983) On estimation of point process intensities, contributions to statistics: essays in honour of Norman L. Johnson, North-Holland, AmsterdamGoogle Scholar
  22. 22.
    Meeker WQ, Escobar LA (1998) Statistical methods for reliability data. Wiley, New YorkMATHGoogle Scholar
  23. 23.
    Nelson W (2003) Recurrent events data analysis for product repairs, disease recurrences, and other applications. (ASA-SIAM series on statistics and applied probability)Google Scholar
  24. 24.
    Pham H, Zhang X (2003) NHPP software reliability and cost models with testing coverage. Eur J Oper Res 145(2):443–454MATHCrossRefGoogle Scholar
  25. 25.
    Phillips MJ (2000) Bootstrap confidence regions for the expected ROCOF of a repairable system. IEEE Trans Reliab 49:204–208CrossRefGoogle Scholar
  26. 26.
    Phillips MJ (2001) Estimation of the expected ROCOF of a repairable system with bootstrap confidence region. Qual Reliab Eng Int 17:159–162CrossRefGoogle Scholar
  27. 27.
    Ramlau-Hansen H (1983) Smoothing counting process intensities by means of kernel functions. Ann Stat 11:453–466MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Rausand M, Høyland A (2004) System reliability theory models, statistical methods, and applications. Wiley, New YorkMATHGoogle Scholar
  29. 29.
    Rigdon SE, Basu AP (2000) Statistical methods for the reliability of repairable systems. Wiley, New YorkMATHGoogle Scholar
  30. 30.
    Rudemo M (1982) Empirical choice of histograms and kernel density estimators. Scand J Stat 9:65–78MATHMathSciNetGoogle Scholar
  31. 31.
    Silverman BW (1986) Density Estimation for Statistics and Data Analysis. Chapman and Hall, LondonMATHGoogle Scholar
  32. 32.
    Wang Z, Wang J, Liang X (2007) Non-parametric estimation for NHPP software reliability models. J Appl Stat 34(1):107–119MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Xie M (1991) Software reliability modelling. World Scientific, SingaporeMATHGoogle Scholar
  34. 34.
    Zielinski JM, Wolfson DB, Nilakantan L, Confavreux C (1993) Isotonic estimation of the intensity of a nonhomogeneous Poisson process: the multiple realization setup. Can J Stat 21(3):257–268MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  • M. Luz Gámiz
    • 1
  • K. B. Kulasekera
    • 2
  • Nikolaos Limnios
    • 3
  • Bo Henry Lindqvist
    • 4
  1. 1.Facultad Ciencias, Depto. Estadistica e Investigacion OperativaUniversidad GranadaGranadaSpain
  2. 2.Department of Mathematical SciencesClemson UniversityClemsonUSA
  3. 3.Centre de Recherches de Royallieu, Laboratoire de Mathématiques AppliquéesUniversité de Technologie de CompiègneCompiègneFrance
  4. 4.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway

Personalised recommendations