Dynamic multivariate quantile inactivity time and applications in investigation of a treatment effect

Mohamed Kayid; Mohamed Kayid

doi:10.3934/math.20241449

AIMS Mathematics

2024, Volume 9, Issue 11: 30000-30014. doi: 10.3934/math.20241449

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Dynamic multivariate quantile inactivity time and applications in investigation of a treatment effect

Mohamed Kayid ^,

Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Received: 08 August 2024 Revised: 27 September 2024 Accepted: 15 October 2024 Published: 22 October 2024

To investigate potentially dependent lifetimes, it is necessary to extend the $ \alpha $-quantile inactivity time to bivariate and multivariate frameworks. To extend this measure to a dynamic multivariate framework, all possible trajectories at time $ t $ are considered. The behavior of the extended $ \alpha $-quantile of inactivity time was investigated in relation to the corresponding multivariate hazard rate function. The $ \alpha $-quantile of the inactivity order is defined and discussed for the multivariate case. The difference between the two bivariate $ \alpha $-quantile functions of inactivity, which is an important measure for studying the effect of treatment on lifespan, was also investigated. This measure was used to analyze the effect of laser treatment on the delay of blindness. Two bootstrap approaches were implemented to construct confidence bounds for the difference measure.
- multivariate quantile inactivity time,
- reversed hazard rate,
- stochastic orders,
- bootstrap method
Citation: Mohamed Kayid. Dynamic multivariate quantile inactivity time and applications in investigation of a treatment effect[J]. AIMS Mathematics, 2024, 9(11): 30000-30014. doi: 10.3934/math.20241449

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Abstract

To investigate potentially dependent lifetimes, it is necessary to extend the $ \alpha $-quantile inactivity time to bivariate and multivariate frameworks. To extend this measure to a dynamic multivariate framework, all possible trajectories at time $ t $ are considered. The behavior of the extended $ \alpha $-quantile of inactivity time was investigated in relation to the corresponding multivariate hazard rate function. The $ \alpha $-quantile of the inactivity order is defined and discussed for the multivariate case. The difference between the two bivariate $ \alpha $-quantile functions of inactivity, which is an important measure for studying the effect of treatment on lifespan, was also investigated. This measure was used to analyze the effect of laser treatment on the delay of blindness. Two bootstrap approaches were implemented to construct confidence bounds for the difference measure.

References

[1]	D. C. Schmittlein, D. G. Morrison, The median residual lifetime: A characterization theorem and an application, Oper. Res., 29 (1981), 392–399. https://doi.org/10.1287/opre.29.2.392 doi: 10.1287/opre.29.2.392
[2]	R. E. Barlow, A. W. Marshall, F. Proschan, Properties of probability distributions with monotone hazard rate, Ann. Math. Statist., 34 (1963), 375–389. https://doi.org/10.1214/aoms/1177704147 doi: 10.1214/aoms/1177704147
[3]	H. W. Block, T. H. Savits, H. Singh, The reversed hazard rate function, Probab. Eng. Inf. Sci., 12 (1998), 69–90. https://doi.org/10.1017/S0269964800005064 doi: 10.1017/S0269964800005064
[4]	A. D. Crescenzo, Some results on the proportional reversed hazards model, Stat. Probab. Lett., 50 (2000), 313–321. https://doi.org/10.1016/S0167-7152(00)00127-9 doi: 10.1016/S0167-7152(00)00127-9
[5]	N. K. Chandra, D. Roy, Some results on reversed hazard rate, Probab. Eng. Inf. Sci., 15 (2001), 95–102. https://doi.org/10.1017/S0269964801151077 doi: 10.1017/S0269964801151077
[6]	M. S. Finkelstein, On the reversed hazard rate, Reliab. Eng. Syst. Saf., 78 (2002), 71–75. https://doi.org/10.1016/S0951-8320(02)00113-8 doi: 10.1016/S0951-8320(02)00113-8
[7]	C. Kundu, A. K. Nanda, T. Hu, A note on reversed hazard rate of order statistics and record values, J. Stat. Plan. Infer., 139 (2009), 1257–1265. https://doi.org/10.1016/j.jspi.2008.08.002 doi: 10.1016/j.jspi.2008.08.002
[8]	X. Li, D. Da, P. Zhao, On reversed hazard rate in general mixture models, Stat. Probab. Lett., 80 (2010), 654–661. https://doi.org/10.1016/j.spl.2009.12.023 doi: 10.1016/j.spl.2009.12.023
[9]	M. Burkschat, N. Torrado, On the reversed hazard rate of sequential order statistics, Stat. Probab. Lett., 85 (2014), 106–113. https://doi.org/10.1016/j.spl.2013.11.015 doi: 10.1016/j.spl.2013.11.015
[10]	M. Esna-Ashari, N. Balakrishnan, M. Alimohammadi, HR and RHR orderings of generalized order statistics, Metrika, 86 (2023), 131–148. https://doi.org/10.1007/s00184-022-00865-2 doi: 10.1007/s00184-022-00865-2
[11]	J. E. Contreras-Reyes, D. I. Gallardo, O. Kharazmi, Time-dependent residual Fisher information and distance for some special continuous distributions, Commun. Stat.-Simul. Comput., 53 (2022), 4331–4351. https://doi.org/10.1080/03610918.2022.2146136 doi: 10.1080/03610918.2022.2146136
[12]	N. Unnikrishnan, B. Vineshkumar, Reversed percentile residual life and related concepts, J. Korean Stat. Soc., 40 (2011), 85–92. https://doi.org/10.1016/j.jkss.2010.06.001 doi: 10.1016/j.jkss.2010.06.001
[13]	M. Mahdy, Further results involving percentile inactivity time order and its inference, Metron, 72 (2014), 269–282. https://doi.org/10.1007/s40300-013-0017-9 doi: 10.1007/s40300-013-0017-9
[14]	M. Shafaei Noughabi, Solving a functional equation and characterizing distributions by quantile past lifetime functions, Econ. Qual. Control, 31 (2016), 55–58. https://doi.org/10.1515/eqc-2015-0017 doi: 10.1515/eqc-2015-0017
[15]	M. Shafaei Noughabi, S. Izadkhah, . On the quantile past lifetime of the components of the parallel systems, Commun. Stat.-Theory Methods, 45 (2016), 2130–2136. https://doi.org/10.1080/03610926.2013.875573 doi: 10.1080/03610926.2013.875573
[16]	L. Balmert, J. H. Jeong, Nonparametric inference on quantile lost lifespan, Biometrics, 73 (2017), 252–259. https://doi.org/10.1111/biom.12555 doi: 10.1111/biom.12555
[17]	L. C. Balmert, R. Li, L. Peng, J. H. Jeong, Quantile regression on inactivity time, Stat. Methods Med. Res., 30 (2021), 1332–1346. https://doi.org/10.1177/0962280221995977 doi: 10.1177/0962280221995977
[18]	M. Kayid, Statistical inference of an α-quantile past lifetime function with applications, AIMS Mathematics, 9 (2024), 15346–15360. https://doi.org/10.3934/math.2024745 doi: 10.3934/math.2024745
[19]	A. P. Basu, Bivariate failure rate, J. Am. Stat. Assoc., 66 (1971), 103–104.
[20]	N. L. Johnson, S. Kotz, A vector multivariate hazard rate, J. Multivar. Anal., 5 (1975), 53–66. https://doi.org/10.1016/0047-259X(75)90055-X doi: 10.1016/0047-259X(75)90055-X
[21]	K. R. Nair, N. U. Nair, Bivariate mean residual life, IEEE Trans. Reliab., 38 (1989), 362–364. https://doi.org/10.1109/24.44183 doi: 10.1109/24.44183
[22]	M. Shaked, J. G. Shanthikumar, Dynamic multivariate mean residual life functions, J. Appl. Probab., 28 (1991), 613–629. https://doi.org/10.2307/3214496 doi: 10.2307/3214496
[23]	M. Shafaei Noughabi, M. Kayid, A. M. Abouammoh, Dynamic multivariate quantile residual life in reliability theory, Math. Probl. Eng., 2018 (2018), 1245656. https://doi.org/10.1155/2018/1245656 doi: 10.1155/2018/1245656
[24]	M. Shafaei Noughabi, M. Kayid, Bivariate quantile residual life: a characterization theorem and statistical properties, Stat. Pap., 60 (2019), 2001–2012. https://doi.org/10.1007/s00362-017-0905-9 doi: 10.1007/s00362-017-0905-9
[25]	M. Kayid, Multivariate mean inactivity time functions with reliability applications, Int. J. Reliab. Appl., 7 (2006), 127–140.
[26]	F. Buono, E. De Santis, M. Longobardi, S. Fabio, Multivariate reversed hazard rates and inactivity times of systems, Methodol. Comput. Appl. Probab., 24 (2022), 1987–2008. https://doi.org/10.1007/s11009-021-09905-2 doi: 10.1007/s11009-021-09905-2
[27]	R. B. Nelsen, Archimedean Copulas, In: An introduction to copulas, New York: Springer, 2006,109–155.
[28]	M. Shaked, J. G. Shanthikumar, Stochastic orders, New York: Springer, 2007. https://doi.org/10.1007/978-0-387-34675-5
[29]	M. Shaked, J. G. Shanthikumar, Multivariate stochastic orderings and positive dependence in reliability theory, Math. Oper. Res., 15 (1990), 545–552. https://doi.org/10.1287/moor.15.3.545 doi: 10.1287/moor.15.3.545
[30]	M. Shaked, J. G. Shanthikumar, Multivariate conditional hazard rate and mean residual life functions and their applications, In: Reliability and Decision Making, New York: Chapman, 1993,137–155.

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