Ordering results for the smallest (largest) and the second smallest (second largest) order statistics of dependent and heterogeneous random variables.

In reliability theory, the kth order statistic represents the lifetime of a (n - k + 1) -out-of-n system. In particular, the smallest (largest) order statistics denotes the lifetime of a series (parallel) system that consist of n components; and the second smallest (second largest) order statistics is the lifetime of a (n - 1) -out-of-n (2-out-of-n) system. In this paper, sufficient conditions are provided to compare the smallest and the second smallest (largest and second largest) order statistics of dependent and heterogeneous random variables having the additive hazard model with the Archimedean copula in the sense of usual stochastic order and hazard rate order. Further, we compare the smallest order statistics of two sets of independent and heterogeneous random variables having the additive hazard model in the sense of dispersive order. Finally, our theoretical findings are evaluated by some numerical examples and counterexamples. [ABSTRACT FROM AUTHOR]