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Aging, Characterization, and Stochastic Ordering

  1. Félix Belzunce1,
  2. Moshe Shaked2

Published Online: 15 JUN 2010

DOI: 10.1002/9780470400531.eorms0014

Wiley Encyclopedia of Operations Research and Management Science

Wiley Encyclopedia of Operations Research and Management Science

How to Cite

Belzunce, F. and Shaked, M. 2010. Aging, Characterization, and Stochastic Ordering. Wiley Encyclopedia of Operations Research and Management Science. .

Author Information

  1. 1

    Universidad de Murcia, Departamento Estadística e Investigación Operativa, Murcia, Spain

  2. 2

    University of Arizona, Department of Mathematics, Tucson, Arizona

Publication History

  1. Published Online: 15 JUN 2010


In this article, we list a number of results that characterize various aging and antiaging notions in terms of stochastic comparisons of related random variables. The random variables that are used in the comparisons are the corresponding residual lives or the corresponding asymptotic equilibrium ages. The aging notions that are studied are the increasing failure rate (IFR), the increasing failure rate average (IFRA), the new better than used (NBU), the decreasing mean residual life (DMRL), as well as other related notions; the corresponding antiaging notions are also characterized in this article. The characterizations are by means of the ordinary stochastic order  ≤ st, of the hazard rate stochastic order  ≤ hr, of the dispersive stochastic order  ≤ disp, of the location-independent riskier stochastic order  ≤ lir, of the excess wealth stochastic order  ≤ ew, of the increasing convex and the increasing concave stochastic orders  ≤ icx and  ≤ icv, of the Laplace transform stochastic order  ≤ Lt, of the likelihood ratio stochastic order  ≤ lr, as well as by means of other related stochastic orders.


  • residual life;
  • asymptotic equilibrium age;
  • increasing failure rate (IFR);
  • increasing failure rate average (IFRA);
  • new better than used (NBU);
  • decreasing mean residual life (DMRL);
  • ordinary stochastic order  ≤ st;
  • hazard rate stochastic order  ≤ hr;
  • dispersive stochastic order  ≤ disp;
  • location-independent riskier stochastic order  ≤ lir;
  • excess wealth stochastic order  ≤ ew;
  • increasing convex stochastic order  ≤ icx;
  • increasing concave stochastic order  ≤ icv;
  • Laplace transform stochastic order  ≤ Lt;
  • likelihood ratio stochastic order  ≤ lr