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Statistical Papers

, Volume 59, Issue 1, pp 199–213 | Cite as

On stochastic comparisons for population age and remaining lifetime

  • Ji Hwan Cha
  • Maxim Finkelstein
Regular Article

Abstract

First, we consider items that are incepted into operation having already a random (initial) age and define the corresponding remaining lifetime. We show that these random variables are identically distributed when the age distribution is equal to the equilibrium distribution of the renewal theory. Then we consider real populations of items that were incepted into operation (were born) at different instants of time and obtain some useful inequalities between the population age and the remaining lifetime using reasoning similar to that employed in population studies. We also discuss the aging properties of populations using different stochastic orders.

Keywords

Equilibrium distribution Random initial age Stable population Stationary population Stochastic ordering 

Notes

Acknowledgments

The authors would like to thank the referee for helpful comments and pointing out the papers dealing with equilibrium distribution. The work of the first author was also supported by Priority Research Centers Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2009-0093827). The work of the second author was supported by the NRF (National Research Foundation of South Africa) grant IFR2011040500026.

References

  1. Arthur WB, Vaupel JW (1984) Some general relationships in population dynamics. Popul Index 50:214–226CrossRefGoogle Scholar
  2. Finkelstein M (2008) Failure rate modeling for reliability and risk. Springer, LondonzbMATHGoogle Scholar
  3. Finkelstein M (2013) On some comparisons of lifetimes for reliability analysis. Reliab Eng Syst Saf 119:300–304CrossRefGoogle Scholar
  4. Finkelstein M, Vaupel JW (2015) On random age and remaining lifetime for population of items. Appl Stoch Model Bus Ind 31:681–689. doi: 10.10002/asmb.2073 MathSciNetCrossRefGoogle Scholar
  5. Glazer RE (1980) Bathtub and related failure rate characterizations. J Am Stat Assoc 76:667–672MathSciNetCrossRefzbMATHGoogle Scholar
  6. Hu T, Kundu A, Nanda AK (2001) On generalized orderings and aging properties with their implications. In: Hayakawa Y, Irony T, Xie M (eds) System and bayesian reliability. World Scientific Printers, SingaporeGoogle Scholar
  7. Keiding N (1990) Statistical inference in the Lexis diagram. Philos Trans R Soc Lond A 332:487–509MathSciNetCrossRefzbMATHGoogle Scholar
  8. Keyfitz N, Casewell N (2005) Applied mathematical demography. Springer, New YorkzbMATHGoogle Scholar
  9. Klugman SA, Panjer HH, Willmot GE (2004) Loss models: from data to decisions, 2nd edn. Wiley, New YorkzbMATHGoogle Scholar
  10. Li X, Xu M (2008) Reversed hazard rate order of equilibrium distributions and a related ageing notion. Stat Pap 49:749–767CrossRefzbMATHGoogle Scholar
  11. Marshall AW, Olkin I (2007) Life distributions. Springer, New YorkzbMATHGoogle Scholar
  12. Preston SH, Heuveline P, Guillot M (2001) Demography: measuring and modeling population processes. Blackwell, New YorkGoogle Scholar
  13. Ross SM (1996) Stochastic processes. John Wiley & Sons, New YorkzbMATHGoogle Scholar
  14. Shaked M, Shanthikumar J (2007) Stochastic orders. Springer, New YorkCrossRefzbMATHGoogle Scholar
  15. Vaupel JW (2009) Life lived and left: carey’s equality. Demogr Res 20(3):7–10CrossRefGoogle Scholar
  16. Whitt W (1985a) The renewal process stationary-excess operator. J Appl Probab 22:156–167MathSciNetCrossRefzbMATHGoogle Scholar
  17. Whitt W (1985b) Uniform conditional variability ordering of probability distributions. J Appl Probab 22:619–633MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of StatisticsEwha Womans UniversitySeoulRepublic of Korea
  2. 2.Department of Mathematical StatisticsUniversity of the Free StateBloemfonteinSouth Africa
  3. 3.ITMO UniversityPetersburgRussia

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