Ordering results of second order statistics from random and non-random number of random variables with Archimedean copulas

In this paper, we investigate stochastic comparisons of the second largest order statistics of homogeneous samples coupled by Archimedean copula, and we establish the reversed hazard rate and likelihood ratio orders, and we further generalize the corresponding results to the case of random sample size. Also, we derive some results for relative ageing between parallel systems and 2-out-of-n/2-out-of-(n + 1) systems. Finally, some examples are given to illustrate the obtained results. Keywords: stochastic orders; archimedean copula; order statistics; random sample size; relative ageing Mathematics Subject Classification: 60E15, 60K10, 90B25