Orderings of the second-largest order statistic with modified proportional reversed hazard rate samples

Order statistics is a significant research topic within probability and statistics, particularly due to its widespread application in areas such as reliability and actuarial science. Extensive research has been conducted on extreme order statistics, and this paper focused on the second-order statistics. Specifically, the study investigated the second-largest order statistics derived from dependent heterogeneous modified proportional reversed hazard rate samples, utilizing the stochastic properties of the Archimedean copula. This paper first examined the usual stochastic order of the second-largest order statistic between two groups of dependent heterogeneous random variables. These variables were analyzed under conditions involving the same tilt parameters with different proportional reversed hazard rate parameters, and different tilt parameters with the same proportional reversed hazard rate parameters. The study derived the sufficient conditions required for establishing the usual stochastic order in these cases. Next, the paper addressed the reversed hazard rate order relationship for the second- largest order statistic between two groups of independent heterogeneous random variables. This analysis was conducted under various conditions: the same tilt parameters with different proportional reversed hazard rate parameters, different tilt parameters with the same proportional reversed hazard rate parameters, and different sample sizes with the same parameters. The sufficient conditions for establishing the reversed hazard rate order were also derived. Finally, the theoretical findings were substantiated through numerical examples, confirming the main conclusions of the paper.