Laplace transform ordering of bivariate inactivity times

Mansour Shrahili; Mohamed Kayid; Mansour Shrahili; Mohamed Kayid

doi:10.3934/math.2022728

AIMS Mathematics

2022, Volume 7, Issue 7: 13208-13224. doi: 10.3934/math.2022728

Previous Article Next Article

Research article

Laplace transform ordering of bivariate inactivity times

Mansour Shrahili ^,,
Mohamed Kayid

Department of Statistics and Operations Research, College of Science, King Saud University, Riyadh 11451, Saudi Arabia

Received: 08 February 2022 Revised: 11 March 2022 Accepted: 15 March 2022 Published: 11 May 2022
MSC : 60E05, 62N05, 60E15

In this paper we consider the Laplace transform of the bivariate inactivity time. We show that a weak bivariate reversed hazard rate order is characterized by the Laplace transform of the bivariate inactivity times in two different frames. The results are used to characterize the weak bivariate reversed hazard rate order using the weak bivariate mean inactivity time order. The results are also used to characterize the decreasing bivariate reversed hazard rate property using the Laplace transform of the bivariate inactivity time.
- bivariate reversed hazard rate order,
- bivariate Laplace transforms order,
- exponential distribution,
- characterization,
- BDRHR
Citation: Mansour Shrahili, Mohamed Kayid. Laplace transform ordering of bivariate inactivity times[J]. AIMS Mathematics, 2022, 7(7): 13208-13224. doi: 10.3934/math.2022728

Related Papers:

Abstract

In this paper we consider the Laplace transform of the bivariate inactivity time. We show that a weak bivariate reversed hazard rate order is characterized by the Laplace transform of the bivariate inactivity times in two different frames. The results are used to characterize the weak bivariate reversed hazard rate order using the weak bivariate mean inactivity time order. The results are also used to characterize the decreasing bivariate reversed hazard rate property using the Laplace transform of the bivariate inactivity time.

References

[1]	I. A. Ahmad, M. Kayid, Characterizations of the RHR and MIT orderings and the DRHR and IMIT classes of life distributions, Prcbab. Eng. Inf. Sci., 19 (2005), 447–461. https://doi.org/10.1017/S026996480505028X doi: 10.1017/S026996480505028X
[2]	R. Ahmadi, Reliability and maintenance modeling for a load-sharing k-out-of-n system subject to hidden failures, Comput. Ind. Eng., 150 (2020), 106894. https://doi.org/10.1016/j.cie.2020.106894 doi: 10.1016/j.cie.2020.106894
[3]	H. Ahmed, M. Kayid, Preservation properties for the Laplace transform ordering of residual lives, Stat. Pap., 45 (2004), 583–590. https://doi.org/10.1007/BF02760570 doi: 10.1007/BF02760570
[4]	F. H. Al-Gashgari, A. I. Shawky, M. A. W. Mahmoud, A nonparametric test for testing exponentiality against NBUCA class of life distributions based on Laplace transform, Qual. Reliab. Eng. Int., 32 (2016), 29–36. https://doi.org/10.1002/qre.1723 doi: 10.1002/qre.1723
[5]	A. Alzaid, J. S. Kim, F. Proschan, Laplace ordering and its applications, J. Appl. Probab., 28 (1991), 116–130. https://doi.org/10.2307/3214745 doi: 10.2307/3214745
[6]	P. Andersen, O. Borgan, R. Gill, N. Keiding, Statistical models based on counting processes, Springer Series in Statistics, 1991. https://doi.org/10.1007/978-1-4612-4348-9
[7]	H. W. Block, T. H. Savits, Burn-in, Stat. Sci., 12 (1997), 1–19. https://doi.org/10.1214/ss/1029963258
[8]	F. Domma, Bivariate reversed hazard rate, notions, and measures of dependence and their relationships, Commun. Stat.-Theor. M., 40 (2011), 989–999. https://doi.org/10.1080/03610920903511777 doi: 10.1080/03610920903511777
[9]	A. Di Crescenzo, P. Di Gironimo, S. Kayal, Analysis of the past lifetime in a replacement model through stochastic comparisons and differential entropy, Mathematics, 8 (2020), 1203. https://doi.org/10.3390/math8081203 doi: 10.3390/math8081203
[10]	L. Eeckhoudt, C. Gollier, Demand for risky assets and the monotone probability ratio order, J. Risk Uncertain., 11 (1995), 113–122. https://doi.org/10.1007/BF01067680 doi: 10.1007/BF01067680
[11]	S. M. El-Arishy, L. S. Diab, E. S. El-Atfy, Characterizations on decreasing Laplace transform of time to failure class and hypotheses testing, Comput. Sci. Comput. Math., 10 (2020), 49–54. https://doi.org/10.20967/jcscm.2020.03.002 doi: 10.20967/jcscm.2020.03.002
[12]	M. Finkelstein, On the reversed hazard rate, Reliab. Eng. Syst. Saf., 78 (2002), 71–75. https://doi.org/10.1016/S0951-8320(02)00113-8
[13]	R. Gupta, A. K. Nanda, Some results on reversed hazard rate ordering, Commun. Stat.-Theor. M., 30 (2001), 2447–2457. https://doi.org10.1081/STA-100107697 doi: 10.1081/STA-100107697
[14]	Z. Guo, J. Zhang, R. Yan, On inactivity times of failed components of coherent system under double monitoring, Prcbab. Eng. Inf. Sci., 2021, 1–18. https://doi.org/10.1017/S0269964821000152
[15]	Y. Jia, J. H. Jeong, Cause-specific quantile regression on inactivity time, Stat. Med., 40 (2021), 1811–1824. https://doi.org/10.1002/sim.8871 doi: 10.1002/sim.8871
[16]	J. Jiang, Z. J. Zhou, X. X. Han, B. C. Zhang, X. D. Ling, A new BRB based method to establish hidden failure prognosis model by using life data and monitoring observation, Knowl. Based Syst., 67 (2014), 270–277. https://doi.org/10.1016/j.knosys.2014.04.045 doi: 10.1016/j.knosys.2014.04.045
[17]	S. Karlin, Total positivity, Stanford University Press, 1968.
[18]	M. Kayid, I. A. Ahmad, On the mean inactivity time ordering with reliability applications, Prcbab. Eng. Inf. Sci., 18 (2004), 395–409. https://doi.org/10.1017/S0269964804183071 doi: 10.1017/S0269964804183071
[19]	M. Kayid, S. Izadkhah, Mean inactivity time function, associated orderings, and classes of life distributions, IEEE Trans. Reliab., 63 (2014), 593–602. https://doi.org/10.1109/TR.2014.2315954 doi: 10.1109/TR.2014.2315954
[20]	M. Kayid, S. Izadkhah, S. Alshami, Laplace transform ordering of time to failure in age replacement models, J. Korean Stat. Soc., 45 (2016), 101–113. https://doi.org/10.1016/j.jkss.2015.08.001 doi: 10.1016/j.jkss.2015.08.001
[21]	N. Keiding, R. Gill, Random truncation models and Markov processes, Ann. Stat., 18 (1990), 582–602. https://doi.org/10.1214/aos/1176347617 doi: 10.1214/aos/1176347617
[22]	N. Keiding, Age-specific incidence and prevalence: A statistical perspective, J. R. Stat. Soc. A Stat., 154 (1991), 371–412. https://doi.org/10.2307/2983150 doi: 10.2307/2983150
[23]	M. Kijima, M. Ohnishi, Stochastic orders and their applications in financial optimization, Math. Methods Oper. Res., 50 (1999), 351–372. https://doi.org/10.1007/s001860050102 doi: 10.1007/s001860050102
[24]	C. Li, X. Li, On stochastic dependence in residual lifetime and inactivity time with some applications, Stat. Probab. Lett., 177 (2021), 109120. https://doi.org/10.1016/j.spl.2021.109120 doi: 10.1016/j.spl.2021.109120
[25]	J. Mulero, F. Pellerey, Bivariate aging properties under Archimedean dependence structures, Commun. Stat.-Theor. M., 39 (2010), 3108–3121. https://doi.org/10.1080/03610920903199987 doi: 10.1080/03610920903199987
[26]	A. K. Nanda, Stochastic orders in terms of Laplace transforms, Bull. Calcutta Stat. Assoc., 45 (1995), 195–202. https://doi.org/10.1177/0008068319950306 doi: 10.1177/0008068319950306
[27]	A. K. Nanda, H. Singh, N. Misra, P. Paul, Reliability properties of reversed residual lifetime, Commun. Stat.-Theor. M., 32 (2003), 2031–2042. https://doi.org/10.1081/STA-120023264 doi: 10.1081/STA-120023264
[28]	E. M. Ortega, A note on some functional relationships involving the mean inactivity time order, IEEE Trans. Reliab., 58 (2008), 172–178. https://doi.org/10.1109/TR.2008.2006576 doi: 10.1109/TR.2008.2006576
[29]	A. Patra, C. Kundu, Further results on residual life and inactivity time at random time, Commun. Stat.-Theor. M., 49 (2020), 1261–1271. https://doi.org/10.1080/03610926.2018.1563170 doi: 10.1080/03610926.2018.1563170
[30]	J. M. Ruiz, J. Navarro, Characterizations based on conditional expectations of the double truncated distribution, Ann. Inst. Stat. Math., 48 (1996), 563–572. https://doi.org/10.1007/BF00050855 doi: 10.1007/BF00050855
[31]	E. Salehi, M. Tavangar, Stochastic comparisons on conditional residual lifetime and inactivity time of coherent systems with exchangeable components, Stat. Probab. Lett., 145 (2019), 327–337. https://doi.org/10.1016/j.spl.2018.10.007 doi: 10.1016/j.spl.2018.10.007
[32]	M. Shaked, T. Wong, Stochastic orders based on ratios of Laplace transforms, J. Appl. Probab., 34 (1997), 404–419. https://doi.org/10.2307/3215380 doi: 10.2307/3215380
[33]	M. Shaked, J. G. Shanthikumar, Stochastic orders, Springer, New York, 2007. https://doi.org/10.1007/978-0-387-34675-5
[34]	T. Tang, D. Lin, D. Banjevic, A. K. Jardine, Availability of a system subject to hidden failure inspected at constant intervals with non-negligible downtime due to inspection and downtime due to repair/replacement, J. Stat. Plan. Infer., 143 (2013), 176–185. https://doi.org/10.1016/j.jspi.2012.05.011 doi: 10.1016/j.jspi.2012.05.011
[35]	C. Tepedelenlioglu, A. Rajan, Y. Zhang, Applications of stochastic ordering to wireless communications, IEEE Trans. Wirel. Commun., 10 (2011), 4249–4257. https://doi.org/10.1109/TWC.2011.093011.110187 doi: 10.1109/TWC.2011.093011.110187
[36]	Y. Wang, H. Pham, A multi-objective optimization of imperfect preventive maintenance policy for dependent competing risk systems with hidden failure, IEEE Trans. Reliab., 60 (2011), 770–781. https://doi.org/10.1109/TR.2011.2167779 doi: 10.1109/TR.2011.2167779
[37]	Y. Zhang, Z. Sun, R. Qin, H. Xiong, Idle duration prediction for manufacturing system using a gaussian mixture model integrated neural network for energy efficiency improvement, IEEE Trans. Autom. Sci. Eng., 18 (2019), 47–55. https://doi.org/10.1109/TASE.2019.2938662 doi: 10.1109/TASE.2019.2938662

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)