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

Mingxia Yang; Mingxia Yang

doi:10.3934/math.2025015

AIMS Mathematics

2025, Volume 10, Issue 1: 311-337. doi: 10.3934/math.2025015

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Research article

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

Mingxia Yang ^,

School of Mathematics and Information Engineering, Longdong University, Qingyang 745000, China

Received: 22 August 2024 Revised: 10 October 2024 Accepted: 16 October 2024 Published: 07 January 2025
MSC : Primary 90B25, Secondary 60E15, 60K10

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.
- Archimedean copula,
- modified proportional reversed hazard rate,
- majorization orders,
- second largest order statistics,
- stochastic orders
Citation: Mingxia Yang. Orderings of the second-largest order statistic with modified proportional reversed hazard rate samples[J]. AIMS Mathematics, 2025, 10(1): 311-337. doi: 10.3934/math.2025015

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Abstract

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.

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