Hostname: page-component-77f85d65b8-grvzd Total loading time: 0 Render date: 2026-04-18T11:42:50.272Z Has data issue: false hasContentIssue false
  • English
  • Français

Ordering and ageing properties of developed sequential order statistics governed by the Archimedean copula

Part of: Distribution theory - Probability Operations research and management science Special processes

Published online by Cambridge University Press:  26 June 2023

Tanmay Sahoo  and
Nil Kamal Hazra
Tanmay Sahoo*
Affiliation:
Indian Institute of Technology Jodhpur
Nil Kamal Hazra*
Affiliation:
Indian Institute of Technology Jodhpur
*
*Postal address: Department of Mathematics, Indian Institute of Technology Jodhpur, Karwar 342037, India.
*Postal address: Department of Mathematics, Indian Institute of Technology Jodhpur, Karwar 342037, India.
Get access Rights & Permissions [Opens in a new window]

Abstract

Developed sequential order statistics (DSOS) are very useful in modeling the lifetimes of systems with dependent components, where the failure of one component affects the performance of remaining surviving components. We study some stochastic comparison results for DSOS in both one-sample and two-sample scenarios. Furthermore, we study various ageing properties of DSOS. We state many useful results for generalized order statistics as well as ordinary order statistics with dependent random variables. At the end, some numerical examples are given to illustrate the proposed results.

Keywords

k-out-of-n systemsstochastic ageingsstochastic orders

MSC classification

Primary: 90B25: Reliability, availability, maintenance, inspection
Secondary: 60E15: Inequalities; stochastic orderings 60K10: Applications (reliability, demand theory, etc.)

Information

Type
Original Article
Information
Advances in Applied Probability , Volume 56 , Issue 2 , June 2024 , pp. 464 - 494
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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.)

Article purchase

Temporarily unavailable

References

Balakrishnan, N., Beutner, E. and Kamps, U. (2008). Order restricted inference for sequential k-out-of-n systems. J. Multivariate Anal. 99, 1489–1502.10.1016/j.jmva.2008.04.014CrossRefGoogle Scholar
Balakrishnan, N. and Zhao, P. (2013). Ordering properties of order statistics from heterogeneous populations: a review with an emphasis on some recent developments. Prob. Eng. Inf. Sci. 27, 403–443.CrossRefGoogle Scholar
Baratnia, M. and Doostparast, M. (2019). Sequential order statistics from dependent random variables. Commun. Statist. Theory Meth. 48, 4569–4580.CrossRefGoogle Scholar
Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rinehart and Winston, New York.Google Scholar
Barmalzan, G., Haidari, A. and Balakrishnan, N. (2018). Univariate and multivariate stochastic orderings of residual lifetimes of live components in sequential $(n-r+ 1)$ -out-of-n systems. J. Appl. Prob. 55, 834–844.CrossRefGoogle Scholar
Belzunce, F., Franco, M., Ruiz, J. M. and Ruiz, M. C. (2001). On partial orderings between coherent systems with different structures. Prob. Eng. Inf. Sci. 15, 273–293.CrossRefGoogle Scholar
Belzunce, F., Mercader, J. A. and Ruiz, J. M. (2003). Multivariate aging properties of epoch times of nonhomogeneous processes. J. Multivariate Anal. 8, 335–350.10.1016/S0047-259X(02)00064-7CrossRefGoogle Scholar
Belzunce, F., Mercader, J. A. and Ruiz, J. M. (2005). Stochastic comparisons of generalized order statistics. Prob. Eng. Inf. Sci. 19, 99–120.CrossRefGoogle Scholar
Block, H. W. and Savits, T. H. (1980). Multivariate increasing failure rate average distributions. Ann. Prob. 8, 793–801.10.1214/aop/1176994666CrossRefGoogle Scholar
Block, H. W., Li, Y. and Savits, T. H. (2003). Preservation of properties under mixture. Prob. Eng. Inf. Sci. 17, 205–212.CrossRefGoogle Scholar
Burkschat, M. (2009). Systems with failure-dependent lifetimes of components. J. Appl. Prob. 46, 1052–1072.10.1239/jap/1261670688CrossRefGoogle Scholar
Burkschat, M., Kamps, U. and Kateri, M. (2010). Sequential order statistics with an order statistics prior. J. Multivariate Anal. 101, 1826–1836.CrossRefGoogle Scholar
Burkschat, M. and Navarro, J. (2011). Aging properties of sequential order statistics. Prob. Eng. Inf. Sci. 25, 449–467.CrossRefGoogle Scholar
Burkschat, M. and Navarro, J. (2013). Dynamic signatures of coherent systems based on sequential order statistics. J. Appl. Prob. 50, 272–287.CrossRefGoogle Scholar
Burkschat, M. and Navarro, J. (2018). Stochastic comparisons of systems based on sequential order statistics via properties of distorted distributions. Prob. Eng. Inf. Sci. 32, 246–274.CrossRefGoogle Scholar
Burkschat, M. and Torrado, N. (2014). On the reversed hazard rate of sequential order statistics. Statist. Prob. Lett. 85, 106–113.CrossRefGoogle Scholar
Cramer, E. and Kamps, U. (1996). Sequential order statistics and k-out-of-n systems with sequentially adjusted failure rates. Ann. Inst. Statist. Math. 48, 535–549.CrossRefGoogle Scholar
Cramer, E. and Kamps, U. (2001). Sequential k-out-of-n systems. In Handbook of Statistics, Vol. 20, Advances in Reliability, eds Balakrishnan, N. and Rao, C. R., North-Holland, Amsterdam, pp. 301–372.Google Scholar
Cramer, E. and Kamps, U. (2003). Marginal distributions of sequential and generalized order statistics. Metrika 58, 293–310.CrossRefGoogle Scholar
Franco, M., Ruiz, J. M. and Ruiz, M. C. (2002). Stochastic orderings between spacings of generalized order statistics. Prob. Eng. Inf. Sci. 16, 471–484.10.1017/S0269964802164059CrossRefGoogle Scholar
Franco, M., Ruiz, M. C. and Ruiz, J. M. (2003). A note on closure of the ILR and DLR classes under formation of coherent systems. Statist. Papers 44, 279–288.CrossRefGoogle Scholar
Hazra, N. K., Kuiti, M. R., Finkelstein, M. and Nanda, A. K. (2017). On stochastic comparisons of maximum order statistics from the location-scale family of distributions. J. Multivariate Anal. 160, 31–41.CrossRefGoogle Scholar
Hazra, N. K., Kuiti, M. R., Finkelstein, M. and Nanda, A. K. (2018). On stochastic comparisons of minimum order statistics from the location-scale family of distributions. Metrika 81, 105–123.CrossRefGoogle Scholar
Hu, T. and Zhuang, W. (2005). A note on stochastic comparisons of generalized order statistics. Statist. Prob. Lett. 72, 163–170.CrossRefGoogle Scholar
Kamps, U. (1995). A concept of generalized order statistics. J. Statist. Planning Infer. 48, 1–23.10.1016/0378-3758(94)00147-NCrossRefGoogle Scholar
Kamps, U. and Cramer, E. (2001). On distributions of generalized order statistics. Statistics 35, 269–280.CrossRefGoogle Scholar
Lai, C. D. and Xie, M. (2006). Stochastic Ageing and Dependence for Reliability. Springer, New York.Google Scholar
Li, X. and Fang, R. (2015). Ordering properties of order statistics from random variables of Archimedean copulas with applications. J. Multivariate Anal. 133, 304–320.CrossRefGoogle Scholar
Marshall, A. W. and Olkin, I. (2007). Life Distributions. Springer, New York.Google Scholar
McNeil, A. J. and Néslehová, J. (2009). Multivariate Archimedeaen copulas, D-monotone functions and $\ell_1$ -norm symmetric distributions. Ann. Statist. 37, 3059–3097.CrossRefGoogle Scholar
Navarro, J. and Burkschat, M. (2011). Coherent systems based on sequential order statistics. Naval Res. Logistics 58, 123–135.CrossRefGoogle Scholar
Navarro, J. (2021). Introduction to System Reliability Theory. Springer, Cham.Google Scholar
Nelsen, R. B. (2007). An Introduction to Copulas. Springer, New York.Google Scholar
sterdal, L. P. (2006). Subadditive functions and their (pseudo-)inverses. J. Math. Anal. Appl. 317, 724–731.CrossRefGoogle Scholar
Samaniego, F. J. (2007). System Signatures and Their Applications in Engineering Reliability. Springer, New York.10.1007/978-0-387-71797-5CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.CrossRefGoogle Scholar
Sahoo, T. and Hazra, N. K. (2023). Ordering and aging properties of systems with dependent components governed by the Archimedean copula. Prob. Eng. Inf. Sci. 37, 1–28.CrossRefGoogle Scholar
Sengupta, D. and Deshpande, J. V. (1994). Some results on the relative ageing of two life distributions. J. Appl. Prob. 31, 991–1003.CrossRefGoogle Scholar
Torrado, N., Lillo, R. E. and Wiper, M. P. (2012). Sequential order statistics: ageing and stochastic orderings. Methodology Comput. Appl. Prob. 14, 579–596.CrossRefGoogle Scholar
Zhuang, W. and Hu, T. (2007). Multivariate stochastic comparisons of sequential order statistics. Prob. Eng. Inf. Sci. 21, 47–66.CrossRefGoogle Scholar