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The second-largest order statistic is of special importance in reliabilitytheory since it represents the time to failure of a $2$-out-of-$n$ system.Consider two $2$-out-of-$n$ systems with heterogeneous random lifetimes. Thelifetimes are assumed to follow heterogeneous general exponentiatedlocation-scale models. In this communication, the usual stochastic and reversedhazard rate orders between the systems' lifetimes are established under twocases. For the case of independent random lifetimes, the usual stochastic orderand the reversed hazard rate order between the second-largest order statisticsare obtained by using the concept of vector majorization and related orders.For the dependent case, the conditions under which the usual stochastic orderbetween the second-largest order statistics holds are investigated. Toillustrate the theoretical findings, some special cases of the exponentiatedlocation-scale model are considered.

Some new ordering results on stochastic comparisons of second largest  order statistics from independent and interdependent heterogeneous  distributions