Modeling bivariate lifetimes based on expected present values of residual lives

Original Paper

DOI: 10.1007/s00477-009-0354-7

Cite this article as:
Ortega, EM. Stoch Environ Res Risk Assess (2010) 24: 675. doi:10.1007/s00477-009-0354-7


This note introduces new concepts of bivariate ageing, that are called the bivariate decreasing (increasing) expected present value residual life, and the bivariate new better (worse) than used expected present value residual life. Interpretations of these notions for the description of the ageing of systems operating under random environments, by means of time-scales and expected present values are given. They can be applied to analyze several biological and engineering systems, from the study of the joint probabilistic behaviour of their correlated components, and to assess the degree or the pattern of their ageing. We provide some examples of bivariate distributions that fulfil these properties, including these conditions for the Marshall–Olkin shock model.


Bivariate exponential Bivariate ageing Marshall–Olkin shock model Present value Joint survival function Laplace order 

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Centro de Investigación Operativa, Departamento Estadística, Matemáticas e InformáticaUniversidad Miguel HernándezOrihuelaSpain

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