Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-27T08:22:05.840Z Has data issue: false hasContentIssue false
  • English
  • Français

On stochastic ordering among extreme shock models

Published online by Cambridge University Press:  28 October 2022

Sirous Fathi Manesh ,
Muhyiddin Izadi  and
Baha-Eldin Khaledi Open the ORCID record for Baha-Eldin Khaledi [Opens in a new window]
Sirous Fathi Manesh
Affiliation:
Department of Statistics, University of Kurdistan, Sanandaj, Iran
Muhyiddin Izadi
Affiliation:
Department of Statistics, Razi University, Kermanshah, Iran
Baha-Eldin Khaledi
Affiliation:
Department of Applied Statistics and Research Methods, University of Northern Colorado, Greeley, CO, USA. E-mail: bahaedin.khaledi@unco.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

In the usual shock models, the shocks arrive from a single source. Bozbulut and Eryilmaz [(2020). Generalized extreme shock models and their applications. Communications in Statistics – Simulation and Computation 49(1): 110–120] introduced two types of extreme shock models when the shocks arrive from one of $m\geq 1$ possible sources. In Model 1, the shocks arrive from different sources over time. In Model 2, initially, the shocks randomly come from one of $m$ sources, and shocks continue to arrive from the same source. In this paper, we prove that the lifetime of Model 1 is less than Model 2 in the usual stochastic ordering. We further show that if the inter-arrival times of shocks have increasing failure rate distributions, then the usual stochastic ordering can be generalized to the hazard rate ordering. We study the stochastic behavior of the lifetime of Model 2 with respect to the severity of shocks using the notion of majorization. We apply the new stochastic ordering results to show that the age replacement policy under Model 1 is more costly than Model 2.

Keywords

Age replacementGeometric distributionHazard rate orderMajorizationUsual stochastic order
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 37 , Issue 4 , October 2023 , pp. 961 - 972
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Belzunce, F., Riquelme, C.M., & Mulero, J (2016). An introduction to stochastic orders. Amsterdam: Academic Press.Google Scholar
Bozbulut, A.A. & Eryilmaz, S. (2020). Generalized extreme shock models and their applications. Communications in Statistics – Simulation and Computation 49(1): 110–120.Google Scholar
Eryilmaz, S. (2017). Computing optimal replacement time and mean residual life in reliability shock models. Computers and Industrial Engineering 103: 40–45.CrossRefGoogle Scholar
Eryilmaz, S. (2017). $\delta$-shock model based on Polya process and its optimal replacement policy. European Journal of Operational Research 263: 690–697.CrossRefGoogle Scholar
Eryilmaz, S. & Devrim, Y. (2019). Reliability and optimal replacement policy for a $k$-out-of-$n$ system subject to shocks. Reliability Engineering & System Safety 188: 393–397.CrossRefGoogle Scholar
Eryilmaz, S. & Kan, C. (2021). A new shock model with a change in shock size distribution. Probability in the Engineering and Informational Sciences 35: 381–395.CrossRefGoogle Scholar
Eryilmaz, S. & Tekin, M. (2019). Reliability evaluation of a system under a mixed shock model. Journal of Computational and Applied Mathematics 352: 255–261.CrossRefGoogle Scholar
Gut, A. (1990). Cumulative shock models. Advances in Applied Probability 22(2): 504–507.CrossRefGoogle Scholar
Gut, A. (1999). Extreme shock models. Extremes 2(3): 295–307.CrossRefGoogle Scholar
Gut, A. (2001). Mixed shock models. Bernoulli 7(3): 541–555.CrossRefGoogle Scholar
Li, Z. & Zhao, P. (2007). Reliability analysis on the $\delta$-shock model of complex systems. IEEE Transactions on Reliability 56(2): 340–348.CrossRefGoogle Scholar
Li, Z., Chan, L.Y., & Yuan, Z. (1999). Failure time distribution under a $\delta$-shock model and its application to economic design of systems. International Journal of Reliability, Quality and Safety Engineering 6: 237–247.CrossRefGoogle Scholar
Lorvand, H., Kelkinnama, M. (2022). Reliability analysis and optimal replacement for a $k$-out-of-$n$ system under a $\delta$-shock model. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability. doi:10.1177/1748006X221082762.Google Scholar
Mallor, F. & Omey, E. (2001). Shocks, runs and random sums. Journal of Applied Probability 38: 438–448.CrossRefGoogle Scholar
Marshall, A.W., Olkin, I., & Arnold, B.C. (2011). Inequalities: theory of majorization and its applications. New York: Springer.CrossRefGoogle Scholar
Nakagawa, T. (2007). Shock and damage models in reliability theory. London: Springer Science and Business Media.Google Scholar
Parvardeh, A. & Balakrishnan, N. (2015). On mixed $\delta$-shock models. Statistics & Probability Letters 102: 51–60.CrossRefGoogle Scholar
Schmidt, K.D. (2014). On inequalities for moments and the covariance of monotone functions. Insurance: Mathematics and Economics 55: 91–95.Google Scholar
Shaked, M. (1982). Dispersive ordering of distributions. Journal of Applied Probability 16: 310–320.CrossRefGoogle Scholar
Shaked, M. & Shanthikumar, J.G. (2007). Stochastic orders. New York: Springer Science and Business Media.CrossRefGoogle Scholar
Shanthikumar, J.G. & Sumita, U. (1983). General shock models associated with correlated renewal sequences. Journal of Applied Probability 20: 600–614.CrossRefGoogle Scholar
Wang, G.J. & Zhang, Y.L. (2005). A shock model with two-type failures and optimal replacement policy. International Journal of Systems Science 36: 209–214.CrossRefGoogle Scholar
Zhao, X., Cai, K., Wang, X., & Song, Y. (2018). Optimal replacement policies for a shock model with a change point. Computers & Industrial Engineering 118: 383–393.CrossRefGoogle Scholar
Zhao, X., Dong, B., Wang, X., & Song, Y. (2022). Reliability assessment for a $k$-out-of-$n$: F system supported by a multi-state protective device in a shock environment. Computers & Industrial Engineering 171. doi:10.1016/j.cie.2022.108426.CrossRefGoogle Scholar