Abstract
In this article, we enhance the study of stochastic comparisons and ageing properties of residual lifetime mixture models. To this aim, first we perform stochastic comparisons of two different mixture models under likelihood ratio, hazard rate, mean residual life and variance residual life orders having different baseline distributions as well as two different mixing distributions. Then, we develop some sufficient conditions which lead to the stochastic comparisons of these mixture models based on reversed hazard rate, mean inactivity time and variance inactivity time orders. Furthermore, we show that increasing likelihood ratio, increasing failure rate, decreasing mean residual life, increasing variance residual life and decreasing variance residual life classes are preserved under the formation of the model. Few applications in reliability engineering are also investigated.
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References
Barlow RE, Proschan F (1981) Statistical theory of reliability and life testing: probability models. To begin with. Silver Spring, Maryland
Belzunce F, Riquelme CM, Mulero J (2015) An introduction to stochastic orders. Academic Press, New York
Block HW, Li Y, Savits TH (2003) Preservaton of properties under mixture. Probab Eng Inf Sci 17(2):205–212
Cai N, Zheng Y (2012) Preservation of generalized ageing classes on the residual life at random time. J Stat Plan Inference 142:148–154
Cha JH, Finkelstein M (2018) On stochastic comparisons for population age and remaining lifetime. Stat Pap 59:199–213
Dewan I, Khaledi BE (2014) On stochastic comparisons of residual life time at random time. Statist Probab Lett 88:73–79
Finkelstein M (2002) Modeling lifetimes with unknown initial age. Appl Stoch Model Bus Ind 76:75–80
Finkelstein M, Vaupel JW (2015) On random age and remaining lifetime for populations of items. Reliab Eng Syst Saf 31:681–689
Gupta RC (2006) Variance residual life function in reliability studies. Metron 3:343–355
Gupta RC, Kirmani SNUA, Launer RL (1987) On life distributions having monotone residual variance. Probab Eng Inf Sci 1(3):299–307
Gupta N, Gandotra N, Bajaj R (2012) Reliability properties of residual lifetime and inactivity time of series and parallel system. J Appl Math Statist Inf 8:5–16
Hazra N, Finkelstein M, Cha JH (2017) Stochastic ordering for populations of manufactured items. TEST 27:173–196
Joag-Dev K, Kochar S, Proschan F (1995) A general composition theorem and its applications to certain partial orderings of distributions. Statist Probab Lett 22:111–119
Karlin S (1968) Total positivity. Stanford University Press, California
Kayid M, Izadkhah S (2015) A new extended mixture model of residual lifetime distributions. Oper Res Lett 43(2):183–188
Kayid M, Izadkhah S (2016) Some new results about the variance inactivity time ordering with applications. Appl Math Model 40:3832–3842
Khaledi BE, Shaked M (2010) Stochastic comparisons of multivariate mixtures. J Multivar Anal 101:2486–2498
Kundu C, Patra A (2018) Some results on residual life and inactivity time at random time. Commun Statist Theory Methods 47:372–384
Li X, Zuo MJ (2004) Stochastic comparison of residual life and inactivity time at a random time. Stoch Model 20:229–235
Li X, Xu M (2006) Some results about MIT order and IMIT class of life distributions. Probab Eng Inf Sci 20(3):481–496
Li X, Fang R (2018) Stochastic properties of two general versions of the residual lifetime at random times. Appl Stoch Model Bus Ind 34:1–16
Lynch JD (1999) On conditions for mixtures of increasing failure rate distributions to have an increasing failure rate. Probab Eng Inf Sci 13(1):33–36
Mahdy M (2012) Characterization and preservations of the variance inactivity time ordering and the increasing variance inactivity time class. J Adv Res 3:29–34
Marshall AW, Olkin I (2007) Life distribution: structure of nonparametric, semi-parametric, and parametric families. Springer, New York
Misra N, Naqvi S (2018a) Some unified results on stochastic properties of residual lifetimes at random times. Braz J Probab Statist 32(2):422–436
Misra N, Naqvi S (2018b) Stochastic comparison of residual lifetime mixture models. Oper Res Lett 46:122–127
Misra N, Gupta N, Dhariyal ID (2008) Stochastic properties of residual life and inactivity time at random time. Stoch Model 24:89–102
Müller A, Stoyan D (2002) Comparison methods for Stochastic models and risks. Wiley, New York
Nanda AK, Kundu A (2009) On generalized stochastic orders of dispersion-type. Calcutta Statist Assoc Bull 61:241–244
Patra A, Kundu C (2019) On generalized orderings and ageing classes for residual life and inactivity time at random time. Metrika 82:691–704
Patra A, Kundu C (2020a) Further results on residual life and inactivity time at random time. Commun Statist Theory Methods 49(5):1261–1271
Patra A, Kundu C (2020b) Stochastic comparisons and ageing properties of RLRT (ITRT) based on variance residual life. Commun Statist Theory Methods. https://doi.org/10.1080/03610926.2020.1812655
Shaked M, Shanthikumar JG (2007) Stochastic orders. Springer, New York
Yue D, Cao J (2000) Some results on the residual life at random time. Acta Math Appl Sin 16:436–443
Acknowledgements
The authors sincerely thank the anonymous Reviewers for constructive comments and suggestions which have considerably improved the content of the manuscript. The financial support (Ref. No. SR/FTP/MS-016/2012) rendered by the Department of Science and Technology, Government of India is gratefully acknowledged.
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Patra, A., Kundu, C. Stochastic comparisons and ageing properties of residual lifetime mixture models. Math Meth Oper Res 94, 123–143 (2021). https://doi.org/10.1007/s00186-021-00750-0
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DOI: https://doi.org/10.1007/s00186-021-00750-0