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Central Limit Theorem for a Family of Reliability Measures

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Recent Advances in Reliability and Quality in Design

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

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Abstract

The main objective of this chapter is to present and prove a Central Limit Theorem for a measure of reliability, called gauge measure, which was introduced in an earlier paper. This measure is derived from a marked point process where the base process is a random point process and the marks are fuzzy random variables. The underlying point process represents the locations where faults are located and the fuzzy marks quantify the subjective assessment when remedial actions are implemented to restore the system to its former working condition. Several examples including an application are provided which serve to illustrate the many results presented in this chapter. Finally, we conclude this chapter with suggestions for future work.

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References

  1. Renyi A (1970) Foundations of probability. Holden-Day, San Francisco

    MATH  Google Scholar 

  2. Davidson J (1994) Stochastic limit theory: an introduction for econometricians. Oxford University Press, Oxford

    Google Scholar 

  3. Billingsley P (1968) Convergence of probability measures. Wiley, New York

    MATH  Google Scholar 

  4. Cressie N (1979) A central limit theorem for random sets. Z. Wahrscheinlichkeitstheorie verw. Gebiete 49:37–47

    Article  MathSciNet  MATH  Google Scholar 

  5. Weil W (1982) An application of the Central Limit for Banach-space-valued random variables to the theory or random sets. Z Wahrscheinlichkeitstheorie verw. Gebiete 60:203–208

    Article  MathSciNet  MATH  Google Scholar 

  6. Jain NC, Marcus MB (1975) Central limit theorem for C(S)-valued random variables. Journal of Functional Analysis 19:216–231

    Article  MathSciNet  MATH  Google Scholar 

  7. Klement EP, Puri ML, Ralescu DA (1986) Limit theorems for fuzzy random variables. Proc R Soc Lond A 407:171–182

    Article  MathSciNet  MATH  Google Scholar 

  8. Radström H (1952) An embedding theorem for spaces of convex sets. Proc Am Math Soc 3:165–169

    Article  MATH  Google Scholar 

  9. Puri ML, Ralescu DA (1985) The concept of normality for fuzzy random variables. Annals of Probability 13:1373–1379

    Article  MathSciNet  MATH  Google Scholar 

  10. Feng Y (2000) Gaussian fuzzy random variables. Fuzzy Sets and Systems 111:325–330

    Article  MathSciNet  MATH  Google Scholar 

  11. Diamond P, Kloeden P (1994) Metric spaces of fuzzy sets. World Scientific, Singapore

    MATH  Google Scholar 

  12. Zeephongsekul P (2006) On a measure of reliability based on point processes with random fuzzy marks. International Journal of Reliability, Quality and Safety Engineering 13:237–255

    Article  Google Scholar 

  13. Snyder DL, Miller MI (1991) Random point processes in time and space, 2nd edn. Springer-Verlag, New York

    MATH  Google Scholar 

  14. Daley DJ, Vere-Jones D (1988) An introduction to the theory of point processes. Springer-Verlag, New York

    MATH  Google Scholar 

  15. Puri ML, Ralescu DA (1978) Fuzzy random variables. J Math Anal Appl 64:409–422

    MathSciNet  Google Scholar 

  16. Kwakernaak H (1978) Fuzzy random variables I. Definitions and theorems. Information Sciences 15:1–29

    Article  MathSciNet  MATH  Google Scholar 

  17. Nahmias S (1978) Fuzzy variables. Fuzzy Sets and Systems 1:97–101

    Article  MathSciNet  MATH  Google Scholar 

  18. Krätschmer V (2001) A unified approach to fuzzy random variables. Fuzzy Sets and Systems 123:1–9

    Article  MathSciNet  MATH  Google Scholar 

  19. Schneider R (1993) Convex bodies: the Brunn–Minkowski theory. Cambridge University Press, Cambridge, UK

    MATH  Google Scholar 

  20. Matheron G (1975) Random sets and integral geometry. Wiley, New York

    MATH  Google Scholar 

  21. Aumann RJ (1965) Integrals of set-valued functions. J Math Anal Appl 12:1–22

    Article  MathSciNet  MATH  Google Scholar 

  22. Körner R (1997) On the variance of fuzzy random variables. Fuzzy Sets and Systems 92:83–93

    Article  MathSciNet  MATH  Google Scholar 

  23. Chung KL (1968) A course in probability theory. Harcourt, Brace & World Inc., New York

    MATH  Google Scholar 

  24. Andrews DWK (1992) Generic uniform convergence. Econometric Theory 8:241–257

    MathSciNet  Google Scholar 

  25. Lehmann EL (1983) Theory of Point Estimation. Springer-Verlag, New York

    MATH  Google Scholar 

  26. Zeephongsekul P (2001) On the variability of fuzzy debugging. Fuzzy Sets and Systems 123:29–38

    Article  MathSciNet  MATH  Google Scholar 

  27. Pham H (2000) Software reliability. Springer-Verlag, Singapore

    MATH  Google Scholar 

  28. Xie M (1991) Software reliability modelling. World Scientific, Singapore

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. School of Mathematical and Geospatial Sciences, RMIT University, GPO Box 2476V, Melbourne, Australia

    P. Zeephongsekul

Authors
  1. P. Zeephongsekul
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Editor information

Editors and Affiliations

  1. Department of Industrial and System Engineering, Rutgers, The State University of New Jersey, 96 Frelinghuysen Road, 08854-8018, Piscataway, USA

    Hoang Pham

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Zeephongsekul, P. (2008). Central Limit Theorem for a Family of Reliability Measures. In: Pham, H. (eds) Recent Advances in Reliability and Quality in Design. Springer Series in Reliability Engineering. Springer, London. https://doi.org/10.1007/978-1-84800-113-8_1

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  • DOI: https://doi.org/10.1007/978-1-84800-113-8_1

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-84800-112-1

  • Online ISBN: 978-1-84800-113-8

  • eBook Packages: EngineeringEngineering (R0)

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