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


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.


Equilibrium distribution Random initial age Stable population Stationary population Stochastic ordering 



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.


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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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