Stochastic Orderings and Ageing Properties of Residual Life Lengths of Live Components in (n-k+1)-Out-Of-n Systems

Suppose that a system consists of n independent and identically distributed components and that the life lengths of the n components are X i , i = 1, …, n . For k ∈ {1, …, n - 1}, let X ( k ) 1 , …, X ( k ) n - k be the residual life lengths of the live components following the k th failure in the system. In this paper we extend various stochastic ordering results presented in Bairamov and Arnold (2008) on the residual life lengths of the live components in an ( n - k + 1)-out-of- n system, and also present a new result concerning the multivariate stochastic ordering of live components in the two-sample situation. Finally, we also characterize exponential distributions under a weaker condition than those introduced in Bairamov and Arnold (2008) and show that some special ageing properties of the original residual life lengths get preserved by residual life lengths.