Abstract
In this paper, the residual probability function is applied to analyze the survival probability of two used components relative to each other in the case when their lifetimes are dependent. The expression of the function by copulas has been derived along with some examples of particular copulas. The behaviour of the residual probability function in terms of the underlying dependence is also discussed. The residual probability order is also considered in the dependent case. In the class of Archimedean survival copulas, we prove that the residual probability order implies the usual stochastic order in the reversed direction, and the hazard rate order concludes the residual probability order.
Highlights
In statistical survival analysis, various measures have been proposed to predict the future events like the time of the failure of a system or the time of the death of a life span
The role of dependence structures between lifetime random variables is extremely important in applied probability literature
We study the link between probability R(t) and the survival copula of ( X, Y )
Summary
Various measures have been proposed to predict the future events like the time of the failure of a system or the time of the death of a life span. The hazard rate and the mean residual life of a lifetime unit are two intrinsic characteristics for analyzing lifetime data In reliability and life testing, the residual life of a unit plays an important role in analysis The role of dependence structures between lifetime random variables is extremely important in applied probability literature Let the random variables X and Y denote the lifetimes of two components of the same system
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