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

TL;DR

This paper investigates stochastic comparison results for the second-largest order statistics in two heterogeneous systems, considering both independent and dependent lifetime models, with applications to reliability theory.

Contribution

It provides new ordering results for second-largest order statistics in heterogeneous systems under independence and dependence, extending existing reliability models.

Findings

Stochastic and reversed hazard rate orders established for independent lifetimes.

Conditions for stochastic order under dependence are identified.

Special cases of exponentiated location-scale models illustrate the theoretical results.

Abstract

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.