Evaluation of P(Xt>Yt ) when both Xt and Yt are residual lifetimes of two systems
Article first published online: 7 JUN 2010
© 2010 The Authors. Statistica Neerlandica © 2010 VVS
Volume 64, Issue 4, pages 460–481, November 2010
How to Cite
Zardasht, V. and Asadi, M. (2010), Evaluation of P(Xt>Yt ) when both Xt and Yt are residual lifetimes of two systems. Statistica Neerlandica, 64: 460–481. doi: 10.1111/j.1467-9574.2010.00464.x
- Issue published online: 7 JUN 2010
- Article first published online: 7 JUN 2010
- Received: May 2009. Revised: March 2010.
- failure rate;
- mean residual lifetime;
- mixture of distributions;
- odds ratio;
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.