Mean residual life of kth records under double monitoring

In the study of reliability of technical systems and shock models, the mean residual life function plays an important role. In this paper, we consider the mean residual life of kth records under double monitoring. We introduce the notion of the mean residual life of kth records under the condition that the mth and (m + 1)st shocks arrived before and after t1, respectively, and the (n + 1)st (1 ≤ m < n) shock arrived after t2 (0 < t1 < t2). We study their respective monotonicity and aging properties. Some stochastic ordering properties are also investigated.