, Volume 17, Issue 2, pp 297–310 | Cite as

Generalized orderings of excess lifetimes of renewal processes

  • Félix Belzunce
  • Asok K. Nanda
  • Eva-María Ortega
  • José M. Ruiz
Original Paper


Several concepts of generalized orderings and generalized ageing classes have been considered in the literature (see Fagiuoli and Pellerey, 1993, Nav Res Logist 40: 829–842). They have been used in reliability, economics and actuarial sciences. These generalized notions provide a knowledge of the intrinsic structure of the ageing and ordering properties of random variables and have become an important tool in applied probability.

In this paper, we provide new results about generalized orderings of excess lifetimes at different times of a renewal process when the underlying distribution belongs to some generalized ageing class. We also derive some interpretations and give some discussions of these results.


Excess lifetime Generalized orderings and ageing classes Renewal processes 

Mathematics Subject Classification (2000)

60E15 60K15 60K10 


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  1. Barlow R, Proschan F (1975) Statistical theory of reliability and life testing: Probability models. Holt, Rinehart and Winston, New York Google Scholar
  2. Belzunce F, Ortega EM, Ruiz JM (1999) The Laplace order and ordering of residual lives. Stat Probab Lett 42:145–156 MATHCrossRefMathSciNetGoogle Scholar
  3. Belzunce F, Ortega EM, Ruiz JM (2001) A note on stochastic comparisons of excess lifetimes of renewal processes. J Appl Probab 38:747–753 MATHCrossRefMathSciNetGoogle Scholar
  4. Cao J, Wang Y (1991) The NBUC and NWUC classes of life distributions. J Appl Probab 28:473–479 MATHCrossRefMathSciNetGoogle Scholar
  5. Chen Y (1994) Classes of life distributions and renewal counting processes. J Appl Probab 31:1110–1115 MATHCrossRefMathSciNetGoogle Scholar
  6. Denuit M, Lefèvre C, Shaked M (1998) The s-convex orders among real random variables, with applications. Math Inequalities Appl 1:585–613 MATHGoogle Scholar
  7. Deshpande JV, Kochar SC, Singh H (1986) Aspects of positive ageing. J Appl Probab 23:748–758 MATHCrossRefMathSciNetGoogle Scholar
  8. Fagiuoli E, Pellerey F (1993) New partial orderings and applications. Nav Res Logist 40:829–842 MATHCrossRefMathSciNetGoogle Scholar
  9. Fagiuoli E, Pellerey F (1994) Preservation of certain classes of life distributions under Poisson shock models. J Appl Probab 31:458–465 MATHCrossRefMathSciNetGoogle Scholar
  10. Hu T, Kundu A, Nanda AK (2001) On generalized orderings and ageing properties with their implications. In: Hayakawa Y, Irony T, Xie M (eds) System and Bayesian reliability. Series on quality, reliability and engineering statistics, vol 5. World Scientific, Singapore, pp 199–228 Google Scholar
  11. Hu T, Ma M, Nanda AK (2004a) Characterizations of generalized ageing classes by the excess lifetime. Southeast Asian Bull Math 28:279–285 MATHMathSciNetGoogle Scholar
  12. Hu T, Nanda AK, Xie H, Zhu Z (2004b) Properties of some stochastic orders: a unified study. Nav Res Logist 51:193–216 MATHCrossRefMathSciNetGoogle Scholar
  13. Kaas R, van Heerwaarden AE, Goovaerts MJ (1994) Ordering of actuarial risks. Caire education series. Brussels Google Scholar
  14. Karlin S (1968) Total positivity. Standford University Press, Standford MATHGoogle Scholar
  15. Li X, Kochar SC (2001) Some new results involving the NBU(2) class of life distributions. J Appl Probab 38:242–246 MATHCrossRefMathSciNetGoogle Scholar
  16. Li X, Li Z, Jing B (2000) Some results about NBUC class of life distributions. Stat Probab Lett 46:229–237 MATHCrossRefMathSciNetGoogle Scholar
  17. Mukherjee SP, Chatterjee A (1992) Stochastic dominance of higher orders and its implications. Commun Stat Theory Methods 21:1977–1986 MATHCrossRefMathSciNetGoogle Scholar
  18. Nanda AK (1997) On improvement and deterioration of a repairable system. IAPQR Trans 22:107–113 MATHMathSciNetGoogle Scholar
  19. Nanda AK, Jain K, Singh H (1996a) On closure of some partial orderings under mixtures. J Appl Probab 33:698–706 MATHCrossRefMathSciNetGoogle Scholar
  20. Nanda AK, Jain K, Singh H (1996b) Properties of moments for s-order equilibrium distributions. J Appl Probab 33:1108–1111 MATHCrossRefMathSciNetGoogle Scholar
  21. Ross S (1996) Stochastic processes, 2nd edn. Wiley, New York MATHGoogle Scholar
  22. Shaked M, Shanthikumar JG (1994) Stochastic orders and their applications. Academic, San Diego MATHGoogle Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2007

Authors and Affiliations

  • Félix Belzunce
    • 1
  • Asok K. Nanda
    • 2
  • Eva-María Ortega
    • 3
  • José M. Ruiz
    • 1
  1. 1.Departamento de Estadística e I.O., Facultad de MatemáticasUniversidad de MurciaEspinardo (Murcia)Spain
  2. 2.Department of MathematicsIndian Institute of Technology KharagpurKharagpurIndia
  3. 3.Centro de Investigación Operativa, Dep. Estadística, Matemática e InformáticaUniversidad Miguel HernándezOrihuela, AlicanteSpain

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