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Evaluation of P(Xt>Yt ) when both Xt and Yt are residual lifetimes of two systems

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  • V. Zardasht
  • M. Asadi

Abstract

Let the random variables X and Y denote the lifetimes of two systems. In reliability theory to compare between the lifetimes of X and Y there are several approaches. Among the most popular methods of comparing the lifetimes are to compare the survival functions, the failure rates and the mean residual lifetime functions of X and Y. Assume that both systems are operating at time t > 0. Then the residual lifetimes of them are Xt=X−t | X>t and Yt=Y−t | Y>t, respectively. In this paper, we introduce, by taking into account the age of systems, a time‐dependent criterion to compare the residual lifetimes of them. In other words, we concentrate on function R(t ):=P(Xt>Yt) which enables one to obtain, at time t, the probability that the residual lifetime Xt is greater than the residual lifetime Yt. It is mentioned, in Brown and Rutemiller (IEEE Transactions on Reliability, 22, 1973) that the probability of type R(t) is important for designing as long‐lived a product as possible. Several properties of R(t) and its connection with well‐known reliability measures are investigated. The estimation of R(t) based on samples from X and Y is also discussed.

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  • V. Zardasht & M. Asadi, 2010. "Evaluation of P(Xt>Yt ) when both Xt and Yt are residual lifetimes of two systems," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 64(4), pages 460-481, November.
    • Handle: RePEc:bla:stanee:v:64:y:2010:i:4:p:460-481
    DOI: 10.1111/j.1467-9574.2010.00464.x
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    References listed on IDEAS

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    1. S. Kirmani & Ramesh Gupta, 2001. "On the Proportional Odds Model in Survival Analysis," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(2), pages 203-216, June.
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    Cited by:

    1. Mhamed Mesfioui & Mohamed Kayid, 2021. "Residual Probability Function for Dependent Lifetimes," Mathematics, MDPI, vol. 9(15), pages 1-13, July.

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