### AIMS Mathematics

2021, Issue 6: 6390-6405. doi: 10.3934/math.2021375
Research article

# Ordering results of second order statistics from random and non-random number of random variables with Archimedean copulas

• Received: 10 January 2021 Accepted: 31 March 2021 Published: 12 April 2021
• MSC : 60E15, 60K10, 90B25

• In this paper, we investigate stochastic comparisons of the second largest order statistics of homogeneous samples coupled by Archimedean copula, and we establish the reversed hazard rate and likelihood ratio orders, and we further generalize the corresponding results to the case of random sample size. Also, we derive some results for relative ageing between parallel systems and $2$-out-of-$n$/$2$-out-of-$(n+1)$ systems. Finally, some examples are given to illustrate the obtained results.

Citation: Bin Lu, Rongfang Yan. Ordering results of second order statistics from random and non-random number of random variables with Archimedean copulas[J]. AIMS Mathematics, 2021, 6(6): 6390-6405. doi: 10.3934/math.2021375

### Related Papers:

• In this paper, we investigate stochastic comparisons of the second largest order statistics of homogeneous samples coupled by Archimedean copula, and we establish the reversed hazard rate and likelihood ratio orders, and we further generalize the corresponding results to the case of random sample size. Also, we derive some results for relative ageing between parallel systems and $2$-out-of-$n$/$2$-out-of-$(n+1)$ systems. Finally, some examples are given to illustrate the obtained results.

 [1] R. E. Barlow, F. Proschan, Statistical theory of reliability and life testing: probability models, Holt Rinehart & Winston of Canada Ltd, 1975. [2] R. Fang, X. H. Li, Advertising a second-price auction, J. Math. Econ., 61 (2015), 246–252. doi: 10.1016/j.jmateco.2015.04.003 [3] P. J. Boland, E. El-Neweihi, F. Proschan, Applications of the hazard rate ordering in reliability and order statistics, J. Appl. Prob., 31 (1994), 180–192. doi: 10.2307/3215245 [4] M. Z. Raqab, W. A. Amin, Some ordering results on order statistics and record values, Iapqr Trans., 21 (1996), 1–8. [5] R. B. Bapat, S. C. Kochar, On likelihood-ratio ordering of order statistics, Linear Algebra Appl., 199 (1994), 281–291. doi: 10.1016/0024-3795(94)90353-0 [6] X. H. Li, R. Fang, Ordering properties of order statistics from random variables of archimedean copulas with applications, J. Multivariate Anal., 133 (2015), 304–320. doi: 10.1016/j.jmva.2014.09.016 [7] R. Fang, X. H. Li, Ordering extremes of interdependent random variables, Commun Stat.-Theory M., 47 (2018), 4187–4201. doi: 10.1080/03610926.2017.1371754 [8] M. Mesfioui, M. Kayid, S. Izadkhah, Stochastic comparisons of order statistics from heterogeneous random variables with archimedean copula, Metrika, 80 (2017), 749–766. doi: 10.1007/s00184-017-0626-z [9] G. Barmalzan, N. Balakrishnan, S. M. Ayat, A. Akrami, Orderings of extremes dependent modified proportional hazard and modified proportional reversed hazard variables under archimedean copula, Commun Stat.-Theory M., 2020, 1–22. [10] G. Pledger, F. Proschan, Comparisons of order statistics and of spacings from heterogeneous distributions, In: Optimizing methods in statistics, Proceedings of a Symposium Held at the Center for Tomorrow, New York: Academic Press, 1971, 89–113. [11] B. E. Khaledi, S. Kochar, Some new results on stochastic comparisons of parallel systems, J. Appl. Probab., 37 (2000), 1123–1128. doi: 10.1239/jap/1014843091 [12] N. Torrado, On magnitude orderings between smallest order statistics from heterogeneous beta distributions, J. Math. Anal. Appl., 426 (2015), 824–838. doi: 10.1016/j.jmaa.2015.02.003 [13] N. Torrado, Tail behaviour of consecutive 2-within-m-out-of-n systems with nonidentical components, Appl. Math. Model., 39 (2015), 4586–4592. doi: 10.1016/j.apm.2014.12.042 [14] A. Arriaza, M. A. Sordo, A. Súarez-Llorens, Comparing residual lives and inactivity times by transform stochastic orders, IEEE T. Reliab., 66 (2017), 366–372. doi: 10.1109/TR.2017.2679158 [15] M. V. Koutras, I. S. Triantafyllou, S. Eryilmaz, Stochastic comparisons between lifetimes of reliability systems with exchangeable components, Methodol. Comput. Appl. Probab., 18 (2016), 1081–1095. doi: 10.1007/s11009-014-9433-4 [16] J. Navarro, Stochastic comparisons of coherent systems, Metrika, 81 (2018), 465–482. doi: 10.1007/s00184-018-0650-7 [17] J. Navarro, Y. del Águila, Stochastic comparisons of distorted distributions, coherent systems and mixtures with ordered components, Metrika, 80 (2017), 627–648. doi: 10.1007/s00184-017-0619-y [18] R. Fang, C. Li, X. H. Li, Ordering results on extremes of scaled random variables with dependence and proportional hazards, Statistics, 52 (2018), 458–478. doi: 10.1080/02331888.2018.1425998 [19] J. Navarro, N. Torrado, Y. del Águila, Comparisons between largest order statistics from multiple-outlier models with dependence, Methodol. Comput. Appl. Probab., 20 (2018), 411–433. doi: 10.1007/s11009-017-9562-7 [20] M. M. Zhang, B. Lu, R. F. Yan, Ordering results of extreme order statistics from dependent and heterogeneous modified proportional (reversed) hazard variables, AIMS Mathematics, 6 (2020), 584–606. [21] J. Navarro, F. Durante, J. Fernández-Sánchez, Connecting copula properties with reliability properties of coherent systems, Appl. Stoch. Model. Bus., 2020, DOI: 10.1002/asmb.2579. [22] J. Navarro, J. Mulero, Comparisons of coherent systems under the time-transformed exponential model, TEST, 29 (2020), 255–281. doi: 10.1007/s11749-019-00656-4 [23] V. V. Kalashnikov, S. T. Rachev, Characterization of queueing models and their stability, Adv. Appl. Prob., 17 (1985), 868–886. doi: 10.2307/1427091 [24] M. Rezaei, B. Gholizadeh, S. Izadkhah, On relative reversed hazard rate order, Commun. Stat.-Theory M., 44 (2015), 300–308. doi: 10.1080/03610926.2012.745559 [25] C. D. Lai, M. Xie, Relative ageing for two parallel systems and related problems, Math. Comput. Model., 38 (2003), 1339–1345. doi: 10.1016/S0895-7177(03)90136-1 [26] C. Li, X. H. Li, Relative ageing of series and parallel systems with statistically independent and heterogeneous component lifetimes, IEEE T. Reliab., 65 (2016), 1014–1021. doi: 10.1109/TR.2015.2512226 [27] W. Y. Ding, Y. Y. Zhang, Relative ageing of series and parallel systems: Effects of dependence and heterogeneity among components, Oper. Res. Lett., 46 (2018), 219–224. doi: 10.1016/j.orl.2018.01.005 [28] D. Sengupta, J. V. Deshpande, Some results on the relative ageing of two life distributions, J. Appl. Prob., 31 (1994), 991–1003. doi: 10.1017/S0021900200099514 [29] N. Misra, J. Francis, Relative ageing of (n-k+1)-out-of-n systems, Stat. Probabil. Lett., 106 (2015), 272–280. doi: 10.1016/j.spl.2015.07.013 [30] N. Misra, J. Francis, Relative aging of (n-k+1)-out-of-n systems based on cumulative hazard and cumulative reversed hazard functions, Nav. Res. Log., 65 (2018), 566–575. doi: 10.1002/nav.21822 [31] N. K. Hazra, N. Misra, On relative ageing of coherent systems with dependent identically distributed components, Adv. Appl. Probab., 52 (2020), 348–376. doi: 10.1017/apr.2019.63 [32] N. K. Hazra, N. Misra, On relative aging comparison of coherent systems with identically distributed components, Probab. Eng. Inform. Sc., 2020, 1–15. [33] P. C. Consul, On the distributions of order statistics for a random sample size, Stat. Neerl., 38 (1984), 249–256. doi: 10.1111/j.1467-9574.1984.tb01115.x [34] M. Shaked, T. Wong, Stochastic orders based on ratios of Laplace transforms, J. Appl. Prob., 34 (1997), 404–419. doi: 10.2307/3215380 [35] M. Shaked, T. Wong, Stochastic comparisons of random minima and maxima, J. Appl. Prob., 34 (1997), 420–425. doi: 10.2307/3215381 [36] X. H. Li, M. J. Zuo, Preservation of stochastic orders for random minima and maxima, with applications, Nav. Res. Log., 51 (2004), 332–344. doi: 10.1002/nav.10122 [37] M. Shaked, G. Shanthikumar, Stochastic orders, New York: Springer, 2007. [38] H. J. Li, X. H. Li, Stochastic orders in reliability and risk, New York: Springer, 2013. [39] F. Belzunce, C. Martinez-Riquelme, J. Mulero, An introduction to stochastic orders, London: Elsevier Academic, 2015. [40] C. D. Lai, M. Xie, Stochastic ageing and dependence for reliability, New York: Springer, 2006. [41] R. B. Nelsen, An introduction to copulas, New York: Springer, 2006. [42] S. Karlin, Total positivity, Vol I, Stanford: Stanford University Press, 1968.
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

2.739 2.4

Article outline

Figures(2)

• On This Site