  The second-largest order statistic is of special importance in reliability
theory 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. The
lifetimes are assumed to follow heterogeneous general exponentiated
location-scale models. In this communication, the usual stochastic and reversed
hazard rate orders between the systems' lifetimes are established under two
cases. For the case of independent random lifetimes, the usual stochastic order
and the reversed hazard rate order between the second-largest order statistics
are obtained by using the concept of vector majorization and related orders.
For the dependent case, the conditions under which the usual stochastic order
between the second-largest order statistics holds are investigated. To
illustrate the theoretical findings, some special cases of the exponentiated
location-scale model are considered.
