, Volume 27, Issue 1, pp 173–196 | Cite as

Stochastic ordering for populations of manufactured items

  • Nil Kamal Hazra
  • Maxim Finkelstein
  • Ji Hwan Cha
Original Paper


We develop a theory of stochastic orders for the age and the residual (remaining) lifetime for populations of manufactured identical items. The obtained theoretical results can be used by manufacturers or users for the justified decisions with respect to, e.g., the increase or decrease in the production rate or with respect to the necessary maintenance actions. Specifically, we show that if the random age of a population is smaller (resp. larger) in some stochastic sense than the defined equilibrium age, then it is also smaller (resp. larger) than the corresponding residual lifetime with respect to different stochastic orders. We discuss various stochastic comparisons between the random age and the residual lifetime for one or more populations. Some ageing properties of the random age and the residual lifetime have also been studied.


Equilibrium distribution Random age Stationary population Stochastic ageing classes Stochastic orders 

Mathematics Subject Classification




The authors would like to thank the associate editor and the anonymous reviewers for their valuable constructive comments which led to an improved version of the manuscript. The first author sincerely acknowledges the financial support from the Claude Leon Foundation, South Africa. The work of the second author was supported by the NRF (National Research Foundation of South Africa) grant IFR 2011040500026. The work of the third author was supported by Priority Research Centers Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2009-0093827). The work of the third author was also supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2016R1A2B2014211).


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Copyright information

© Sociedad de Estadística e Investigación Operativa 2017

Authors and Affiliations

  1. 1.Department of Mathematical Statistics and Actuarial ScienceUniversity of the Free StateBloemfonteinSouth Africa
  2. 2.ITMO UniversitySt. PetersburgRussia
  3. 3.Department of StatisticsEwha Womans UniversitySeoulRepublic of Korea

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