Stochastic orderings and ageing properties of residual life lengths of live components in (n-k+1)-out-of-n systems

Abstract

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 kth 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.