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Toward the evaluation of \(P(X_{(t)} > Y_{(t)})\) when both \(X_{(t)}\) and \(Y_{(t)}\) are inactivity times of two systems. (English) Zbl 1508.62230

Summary: The inactivity time, also known as reversed residual life, has been a topic of increasing interest in the literature. In this investigation, based on the comparison of inactivity times of two devices, we introduce and study a new estimate of the probability of the inactivity time of one device exceeding that of another device. The problem studied in this paper is important for engineers and system designers. It would enable them to compare the inactivity times of the products and, hence to design better products. Several properties of this probability are established. Connections between the target probability and the reversed hazard rates of the two devices are established. In addition, some of the reliability properties of the new concept are investigated extending the well-known probability ordering. Finally, to illustrate the introduced concepts, many examples and applications in the context of reliability theory are included.

MSC:

62N05 Reliability and life testing
60E15 Inequalities; stochastic orderings
90B25 Reliability, availability, maintenance, inspection in operations research
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[1] Ahmad, I. A., and M. Kayid. 2005. Characterizations of the RHR and MIT orderings and the DRHR and IMIT classes of life distributions. Probability Eng. Inf. Sci. 19:447-61. · Zbl 1336.60027 · doi:10.1017/S026996480505028X
[2] Ahmad, I. A., M. Kayid, and F. Pellerey. 2005. Further results involving the MIT order and the IMIT class. Probability in the Engineering and Informational Sciences 19:377-95. · Zbl 1075.60010 · doi:10.1017/S0269964805050229
[3] Andersen, P. K., O. Borgan, R. D. Gill, and N. Keiding. 1993. Statistical methods based on counting processes. New York: Springer-Verlag. · Zbl 0769.62061 · doi:10.1007/978-1-4612-4348-9
[4] Bennett, S. 1983. Analysis of survival data by the proportional odd model. Statistics in Medicine 2:273-77. · doi:10.1002/sim.4780020223
[5] Block, H., and T. Savits. 1997. Burn-in. Statistical Science 12:1-13. · doi:10.1214/ss/1029963258
[6] Block, H. W., T. H. Savits, and H. Singh. 1998. The reversed hazard rate function. Probability in the Engineering and Informational Sciences 12:69-70. · Zbl 0972.90018 · doi:10.1017/S0269964800005064
[7] Cha, J., and J. Mi. 2007. Some probability function in reliability and their applications. Naval Research Logistics 54:128-35. · Zbl 1126.62093 · doi:10.1002/nav.20192
[8] Chandra, N. K., and D. Roy. 2001. Some results on reversed hazard rate. Probability in the Engineering and Informational Sciences 15:95-102. · Zbl 1087.62510 · doi:10.1017/S0269964801151077
[9] Di Crescenzo, A. 2000. Some results on the proportional reversed hazards model. Statistics & Probability Letters 50:313-21. · Zbl 0967.60016 · doi:10.1016/S0167-7152(00)00127-9
[10] Eeckhoudt, L., and C. Gollier. 1995. Demand for risky assets and the monotone probability ratio order. Journal of Risk and Uncertainty 11:113-22. · Zbl 0863.90043 · doi:10.1007/BF01067680
[11] Eryilmaz, S. 2010. Mean residual and mean past lifetime of multi-state systems with identical components. IEEE Transactions on Reliability 59:644-49. · doi:10.1109/TR.2010.2054173
[12] Finkelstein, M. S. 2002. On the reversed hazard rate. Reliability Engineering & System Safety 78:71-75. · doi:10.1016/S0951-8320(02)00113-8
[13] Goliforushani, S., M. Asadi, and N. Balakrishnan. 2012. On the residual and inactivity times of the components of used coherent systems. Journal of Applied Probability 49:385-404. · Zbl 1244.90074 · doi:10.1239/jap/1339878793
[14] Hollander, M., and F. J. Samaniego. 2008. The use of stochastic precedence in the comparison of engineered systems. Advances in Mathematical Modeling for Reliability, IOS, Amsterdam, 129-37.
[15] Kalbfleisch, J. D., and J. F. Lawless. 1991. Regression models for right truncated data with applications to AIDS incubation times and reporting lags. Statistica Sinica 1:19-32. · Zbl 0826.62089
[16] Kayid, M., and I. A. Ahmad. 2004. On the mean inactivity time ordering with reliability applications. Probability in the Engineering and Informational Sciences 18:395-409. · Zbl 1059.62105 · doi:10.1017/S0269964804183071
[17] Khanjari, M. S. 2008. Mean past and mean residual life functions of a parallel system with nonidentical components. Communications in Statistics - Theory and Methods 37:1134-45. · Zbl 1138.62065
[18] Kijima, M., and M. Ohnishi. 1999. Stochastic orders and their applications in financial optimization. Mathematical Methods of Operations Research 50:351-72. · Zbl 0958.91020 · doi:10.1007/s001860050102
[19] Kirmani, S., and R. Gupta. 2001. On the proportional odds model in survival analysis. Annals of the Institute of Statistical Mathematics 53:203-16. · Zbl 1027.62072 · doi:10.1023/A:1012458303498
[20] Li, X., and J. Lu. 2003. Stochastic comparisons on residual life and inactivity time of series and parallel systems. Probability in the Engineering and Informational Sciences 17:267-75. · Zbl 1037.60017 · doi:10.1017/S0269964803172087
[21] Li, X., and M. Xu. 2006. Some results about MIT order and IMIT class of life distributions. Probability in the Engineering and Informational Sciences 20:481. · Zbl 1122.60018 · doi:10.1017/S0269964806060293
[22] Marshall, A. W., and I. Olkin. 2007. Life distributions, structure of non-parametric, semi-parametric, and parametric families. New York, USA: Springer, Verlag. · Zbl 1304.62019
[23] Mi, J. 1999. Optimal active redundancy allocation in K-out-of-n system. Journal of Applied Probability 36:927-33. · Zbl 0947.90034 · doi:10.1239/jap/1032374645
[24] Nanda, A. K., H. Singh, N. Misra, and P. Paul. 2003. Reliability properties of reversed residual lifetime. Communications in Statistics - Theory and Methods 32:2031-42. · Zbl 1156.62360
[25] Ortega, E. M. 2009. A note on some functional relationships involving the mean inactivity time order. IEEE Transactions on Reliability 58:172-78. · doi:10.1109/TR.2008.2006576
[26] Rao, N. S. 1997. Distributed decision fusion using empirical estimation. IEEE Transactions on Aerospace and Electronic Systems 33:1106-14. · doi:10.1109/7.624346
[27] Shaked, M., and J. G. Shanthikumar. 2007. Stochastic order. New York, USA: Springer, Verlag. · Zbl 1111.62016 · doi:10.1007/978-0-387-34675-5
[28] Zardasht, V., and M. Asadi. 2010. Evaluation of \(P(X_t > Y_t)\) when both \(X_t\) and \(Y_t\) are residual lifetimes of two systems. Statistical Neerlandica 64:460-81. · doi:10.1111/j.1467-9574.2010.00464.x
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