Ageing Concepts

  • N. Unnikrishnan Nair
  • P. G. Sankaran
  • N. Balakrishnan
Part of the Statistics for Industry and Technology book series (SIT)


A considerable part of reliability theory is dedicated to the study of ageing concepts, their properties, implications and applications. In this chapter, we review some of the important results in this area and translate the basic definitions to make them amenable for a quantile-based analysis. Ageing represents the phenomenon by which the residual life of a unit is affected by its age in some probabilistic sense. It can be positive ageing, negative ageing or no ageing, according to whether the residual lifetime decreases, increases or remains the same as age advances. Generally, one investigates whether a given ageing concept preserves certain reliability operations such as formation of coherent structures, mixtures and convolutions. We first introduce the basic ideas behind convergence, mixtures, convolutions, shock models and equilibrium distributions. The ageing concepts are studied under three broad categories—based on hazard functions, residual life functions and survival functions. The IHR, IHR(2), IGHR, NBUHR, NBUHRA, SIHR, IHRA, DMTTF, IHRA* t 0 classes and their duals along with their properties come under ageing notions related to the hazard function. In the class of concepts based on residual life, we discuss DMRL, DMRLHA, UBA, UBAE, HUBAE, DRMRL, DVRL, DVRLA, NDMRL, NDVRL, IPRL-α, DMERL classes and their duals. Those defined in terms of the survival function include NBU, NBU-t 0, NBU* t 0, NBU(2), SNBU, NBUE, NBU(2)-t 0, NBUL, NBUP-α, NBUE, HNBUE, \(\mathcal{L}\)-class, \(\mathcal{M}\)-class and the renewal notions NBRU, RNBU, RNBUE, RNBRU and RNBRUE. A brief discussion is also made on classes of distributions possessing monotonic properties for reliability concepts in reversed time. The ageing properties of the quantile function models introduced in  Chap. 3 are presented. Finally, some definitions and results on relative ageing are detailed.


Hazard Rate Survival Function Residual Life Quantile Function Coherent System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • N. Unnikrishnan Nair
    • 1
  • P. G. Sankaran
    • 1
  • N. Balakrishnan
    • 2
  1. 1.Department of StatisticsCochin University of Science and TechnologyCochinIndia
  2. 2.Department of Mathematics and StatisticsMcMasters UniversityHamiltonCanada

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