Ordering results of extreme order statistics from dependent and heterogeneous modified proportional (reversed) hazard variables

Miaomiao Zhang; Bin Lu; Rongfang Yan; Miaomiao Zhang; Bin Lu; Rongfang Yan

doi:10.3934/math.2021036

AIMS Mathematics

2021, Volume 6, Issue 1: 584-606. doi: 10.3934/math.2021036

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Research article

Ordering results of extreme order statistics from dependent and heterogeneous modified proportional (reversed) hazard variables

College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

Received: 26 August 2020 Accepted: 08 October 2020 Published: 22 October 2020
MSC : Primary 90B25; Secondary 60E15, 60K10

In this paper, we carry out stochastic comparisons on extreme order statistics (i.e. smallest and largest order statistics) from dependent and heterogeneous samples following modified proportional hazard rates (MPHR) and modified proportional reversed hazard rates (MPRHR) models. We build the usual stochastic order for sample minimums and maximums, and the hazard rate order on minimums of sample and the reversed hazard rate order on maximums of sample are also derived, respectively. Finally, some examples are given to illustrate the theoretical results.
- MPHR and MPRHR models,
- Archimedean copula,
- stochastic orders,
- majorization
Citation: Miaomiao Zhang, Bin Lu, Rongfang Yan. Ordering results of extreme order statistics from dependent and heterogeneous modified proportional (reversed) hazard variables[J]. AIMS Mathematics, 2021, 6(1): 584-606. doi: 10.3934/math.2021036

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Abstract

In this paper, we carry out stochastic comparisons on extreme order statistics (i.e. smallest and largest order statistics) from dependent and heterogeneous samples following modified proportional hazard rates (MPHR) and modified proportional reversed hazard rates (MPRHR) models. We build the usual stochastic order for sample minimums and maximums, and the hazard rate order on minimums of sample and the reversed hazard rate order on maximums of sample are also derived, respectively. Finally, some examples are given to illustrate the theoretical results.

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