Skip to main content

Concepts of Stochastic Dependence in Reliability Analysis

  • Chapter

This is a preview of subscription content, log in via an institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Hutchinson TP, Lai CD. Continuous bivariate distributions, emphasising applications. Adelaide, Australia: Rumsby Scientific Publishing; 1990.

    Google Scholar 

  2. Joe H. Multivariate models and dependence concepts. London: Chapman and Hall; 1997.

    Google Scholar 

  3. Barlow RE, Proschan F. Statistical theory of reliability and life testing: probability models. Silver Spring (MD): To Begin With; 1981.

    Google Scholar 

  4. Lehmann EL. Some concepts of dependence. Ann Math Stat 1966;37:1137–53.

    MATH  MathSciNet  Google Scholar 

  5. Esary JD, Proschan F, Walkup DW. Association of random variables, with applications. Ann Math Stat 1967;38:1466–74.

    MathSciNet  Google Scholar 

  6. Esary JD, Proschan F. Relationships among some bivariate dependence. Ann Math Stat 1972;43:651–5.

    MathSciNet  Google Scholar 

  7. Harris R. A multivariate definition for increasing hazard rate distribution. Ann Math Stat 1970;41:713–7

    MATH  Google Scholar 

  8. Brindley EC, Thompson WA. Dependence and aging aspects of multivariate survival. J Am Stat Assoc 1972;67:822–30.

    MathSciNet  Google Scholar 

  9. Yanagimoto T. Families of positive random variables. Ann Inst Stat Math 1972;26:559–73.

    MathSciNet  Google Scholar 

  10. Shaked M. A concept of positive dependence for exchangeable random variables. Ann Stat 1977;5:505–15.

    MATH  MathSciNet  Google Scholar 

  11. Shaked M. Some concepts of positive dependence for bivariate interchangeable distributions. Ann Inst Stat Math 1979;31:67–84.

    MATH  MathSciNet  Google Scholar 

  12. Shaked M. A general theory of some positive dependence notions. J Multivar Anal 1982;12: 199–218.

    Article  MATH  MathSciNet  Google Scholar 

  13. Block HW, Ting, ML. Some concepts of multivariate dependence. Commun Stat A: Theor Methods 1981;10:749–62.

    MathSciNet  Google Scholar 

  14. Jogdeo K. Dependence concepts and probability inequalities. In: Patil GP, Kotz S, Ord JK, editors. A modern course on distributions in scientific work — models and structures, vol. 1. Dordrecht: Reidel; 1975. p.271–9.

    Google Scholar 

  15. Jogdeo K. Dependence concepts of. In: Encyclopedia of statistical sciences, vol. 2. New York: Wiley; 1982. p.324–34.

    Google Scholar 

  16. Lai CD, Xie M. A new family of positive dependence bivariate distributions. Stat Probab Lett 2000;46:359–64.

    Article  MathSciNet  Google Scholar 

  17. Robbins H. A remark on the joint distribution of cumulative sums. Ann Math Stat 1954;25:614–6.

    MATH  MathSciNet  Google Scholar 

  18. Kimball AW. On dependent tests of significance in analysis of variance. Ann Math Stat 1951;22:600–2.

    MATH  MathSciNet  Google Scholar 

  19. Douglas R, Fienberg SE, Lee MLT, Sampson AR, Whitaker LR. Positive dependence concepts for ordinal contingency tables. In: IMS lecture notes monograph series: topics in statistical dependence, vol. 16. Hayward (CA): Institute of Mathematical Statistics; 1990. p.189–202.

    Google Scholar 

  20. Xie M, Lai, CD. On the increase of the expected lifetime by parallel redundancy. Asia-Pac J Oper Res 1996;13:171–9.

    MathSciNet  Google Scholar 

  21. Mi J. Bolstering components for maximizing system lifetime. Nav Res Logist 1998;45:497–509.

    MATH  MathSciNet  Google Scholar 

  22. Kotz S, Lai CD, Xie, M. The expected lifetime when adding redundancy in systems with dependent components. IIE Trans 2003;in press.

    Google Scholar 

  23. Farlie DJG. The performance of some correlation coefficients for a general bivariate distribution. Biometrika 1960;47:307–23.

    MATH  MathSciNet  Google Scholar 

  24. Johnson NL, Kotz S. Distributions in statistics: continuous multivariate distributions. New York: Wiley; 1972.

    Google Scholar 

  25. Mukerjee SP, Sasmal BC. Life distributions of coherent dependent systems. Calcutta Stat Assoc Bull 1977;26:39–52.

    Google Scholar 

  26. Lingappaiah GS. Bivariate gamma distribution as a life test model. Aplik Mat 1983;29:182–8.

    MathSciNet  Google Scholar 

  27. Philips MJ. A preventive maintenance plan for a system subject to revealed and unrevealed faults. Reliab Eng 1981;2:221–31.

    Google Scholar 

  28. Kotz S, Johnson NL. Some replacement-times distributions in two-component systems. Reliab Eng 1984;7:151–7.

    Google Scholar 

  29. Mardia KV. Families of bivariate distributions. London: Griffin; 1970.

    Google Scholar 

  30. Lindley DV, Singpurwalla ND. Multivariate distributions for the life lengths of components of a system sharing a common environment. J Appl Probab 1986;23:418–31.

    MathSciNet  Google Scholar 

  31. Nayak TK. Multivariate Lomax distribution: properties and usefulness in reliability theory. J Appl Probab 1987;24:170–7.

    MATH  MathSciNet  Google Scholar 

  32. Sankaran PG, Nair NU. A bivariate Pareto model and its applications to reliability. Nav Res Logist 1993;40:1013–20.

    Google Scholar 

  33. Marshall AW, Olkin I. A multivariate exponential distribution. J Am Stat Assoc 1967;62:30–44.

    MathSciNet  Google Scholar 

  34. Block HW, Basu AP. A continuous bivariate exponential distribution. J Am Stat Assoc 1976;64:1031–7.

    MathSciNet  Google Scholar 

  35. Freund J. A bivariate extension of the exponential distribution. J Am Stat Assoc 1961;56:971–7.

    MATH  MathSciNet  Google Scholar 

  36. Lai CD, Moore T. Probability integrals of a bivariate gamma distribution. J Stat Comput Simul 1984;19:205–13.

    MathSciNet  Google Scholar 

  37. Downton F. Bivariate exponential distributions in reliability theory. J R Stat Soc Ser B 1970;32:408–17.

    MATH  MathSciNet  Google Scholar 

  38. Sarmanov OV. Generalized normal correlation and two-dimensional Frechet classes. Dokl Sov Math 1966;168:596–9.

    MathSciNet  Google Scholar 

  39. Lee MLT. Properties and applications of the Sarmanov family of bivariate distributions. Commun Stat A: Theor Methods 1996;25:1207–22.

    MATH  Google Scholar 

  40. Gupta SS. Probability integrals of multivariate normal and multivariate t. Ann Math Stat 1963;34:792–828.

    MATH  Google Scholar 

  41. Nelsen RB. An introduction copulas. Lecture notes in statistics, vol. 139,: New York: Springer-Verlag; 1999.

    Google Scholar 

  42. Bairamov I, Lai CD, Xie M. Bivariate Lomax distribution and generalized Ali-Mikhail-Haq distribution. Unpublished results.

    Google Scholar 

  43. Morgenstern D. Einfache Beispiele zweidimensionaler Verteilungen. Mitteilungsbl Math Stat 1956;8:234–5.

    MATH  MathSciNet  Google Scholar 

  44. Gumbel EJ. Bivariate exponential distributions. J Am Stat Assoc 1960;55:698–707.

    MATH  MathSciNet  Google Scholar 

  45. Huang JS, Kotz S. Modifications of the Farlie-Gumbel-Morgenstern distributions. A tough hill to climb. Metrika 1999;49:135–45.

    MathSciNet  Google Scholar 

  46. Woodworth GG. On the asymptotic theory of tests of independence based on bivariate layer ranks. Technical Report No 75, Department of Statistics, University of Minnesota. See also Abstr Ann Math Stat 1966;36:1609.

    Google Scholar 

  47. Bairamov I, Kotz S. Dependence structure and symmetry of Huang-Kotz FGM distributions and their extensions. Metrika 2002;in press.

    Google Scholar 

  48. Bairamov I, Kotz S, Bekci M. New generalized Farlie-Gumbel-Morgenstern distributions and concomitants of order statistics. J Appl Stat 2001;28:521–36.

    MathSciNet  Google Scholar 

  49. Bairamov I, Kotz S. On a new family of positive quadrant dependent bivariate distributions. GWU/IRRA/TR No 2000/05. The George Washington University, 2001.

    Google Scholar 

  50. Abdel-Hameed M, Sampson AR. Positive dependence of the bivariate and trivariate absolute normal t, X2 and F distributions. Ann Stat 1978;6:1360–8.

    MathSciNet  Google Scholar 

  51. Rödel E. A necessary condition for positive dependence. Statistics 1987;18:351–9.

    MATH  MathSciNet  Google Scholar 

  52. Lai CD, Xie M, Bairamov I. Dependence and ageing properties of bivariate Lomax distribution. In: Hayakawa Y, Irony T, Xie M, editors. A volume in honor of Professor R. E. Barlow on his 70th birthday. Singapore: WSP; 2001. p.243–55.

    Google Scholar 

  53. Kotz S, Balakrishnan N, Johnson NL. Continuous multivariate distributions, vol. 1: models and applications. New York: Wiley; 2000.

    Google Scholar 

  54. Kimeldorf G, Sampson AR. Positive dependence orderings. Ann Inst Stat Math 1987;39:113–28.

    MathSciNet  Google Scholar 

  55. Tchen A. Inequalities for distributions with given marginals. Ann Probab 1980;8:814–27.

    MATH  MathSciNet  Google Scholar 

  56. Shaked M, Shantikumar JG, editors. Stochastic orders and their applications. New York: Academic Press; 1994.

    Google Scholar 

Download references

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer-Verlag London Limited

About this chapter

Cite this chapter

Lai, C.D., Xie, M. (2003). Concepts of Stochastic Dependence in Reliability Analysis. In: Pham, H. (eds) Handbook of Reliability Engineering. Springer, London. https://doi.org/10.1007/1-85233-841-5_7

Download citation

  • DOI: https://doi.org/10.1007/1-85233-841-5_7

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-85233-453-6

  • Online ISBN: 978-1-85233-841-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics