Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T15:42:21.438Z Has data issue: false hasContentIssue false
  • English
  • FranΓ§ais

Univariate and multivariate stochastic orderings of residual lifetimes of live components in sequential (𝑛-π‘Ÿ+ 1)-out-of-𝑛 systems

Part of: Structure and classification of infinite or finite groups Operations research and management science

Published online by Cambridge University Press:Β  16 November 2018

Ghobad Barmalzan ,
Abedin Haidari  and
Narayanaswamy Balakrishnan
Ghobad Barmalzan*
Affiliation:
University of Zabol
Abedin Haidari*
Affiliation:
Shahid Beheshti University
Narayanaswamy Balakrishnan*
Affiliation:
McMaster University
*
* Postal address: Department of Statistics, University of Zabol, Sistan and Baluchestan, Iran. Email address: ghobad.barmalzan@gmail.com
** Postal address: Department of Mathematical Sciences, Shahid Beheshti University, G.C. Evin, Tehran, Iran.
*** Postal address: Department of Mathematics and Statistics, McMaster University, Hamilton, Canada.
Get access Rights & Permissions [Opens in a new window]

Abstract

Sequential order statistics can be used to describe the ordered lifetimes of components of a system when the failure of a component may affect the reliability of the remaining components. After a reliability system consisting of n components fails, some of its components may still be alive. In this paper we first establish some univariate stochastic orderings and ageing properties of the residual lifetimes of the live components in a sequential (n-r+1)-out-of-n system. We also obtain a characterizing result for the exponential distribution based on uncorrelated residual lifetimes of live components. Finally, we provide some sufficient conditions for comparing vectors of residual lifetimes of the live components from two sequential (n-r+1)-out-of-n systems. The results established here extend some well-known results in the literature.

Keywords

Sequential (𝑛-π‘Ÿ+ 1)-out-of-𝑛 systemcharacterizationlikelihood ratio orderhazard rate ordermean residual life orderusual multivariate stochastic orderlive componentresidual lifetime

MSC classification

Primary: 20E15: Chains and lattices of subgroups, subnormal subgroups
Secondary: 90B25: Reliability, availability, maintenance, inspection
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 3 , September 2018 , pp. 834 - 844
Copyright
Copyright Β© Applied Probability Trust 2018Β 

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

Arnold, B. C., Balakrishnan, N. and Nagaraja, H. N. (1992). A First Course in Order Statistics. John Wiley, New York.Google Scholar
Bairamov, I. and Arnold, B. C. (2008). On the residual lifelengths of the remaining components in an n-k+1 out of n system. Statist. Prob. Lett. 78, 945–952.Google Scholar
Balakrishnan, N. and Rao, C. R. (eds) (1998a). Order Statistics: Theory and Methods (Handbook Statist. 16). North-Holland, Amsterdam.Google Scholar
Balakrishnan, N. and Rao, C. R. (eds) (1998b). Order Statistics: Applications (Handbook Statist. 17). North-Holland, Amsterdam.Google Scholar
Balakrishnan, N., Barmalzan, G. and Haidari, A. (2014). Stochastic orderings and ageing properties of residual life lengths of live components in (n-k+1)-out-of-n systems. J. Appl. Prob. 51, 58–68.Google Scholar
Balakrishnan, N., Beutner, E. and Kamps, U. (2008). Order restricted inference for sequential k-out-of-n systems. J. Multivariate Anal. 99, 1489–1502.Google Scholar
Balakrishnan, N., Beutner, E. and Kamps, U. (2011). Modeling parameters of a load-sharing system through link functions in sequential order statistics models and associated inference. IEEE Trans. Reliab. 60, 605–611.Google Scholar
Balakrishnan, N. et al. (2015). Reliability inference on composite dynamic systems based on Burr type-XII distribution. IEEE Trans. Reliab. 64, 144–153.Google Scholar
Belzunce, F., Mercader, J.-A., Ruiz, J.-M. and Spizzichino, F. (2009). Stochastic comparisons of multivariate mixture models. J. Multivariate Anal. 100, 1657–1669.Google Scholar
Burkschat, M. and Navarro, J. (2011). Aging properties of sequential order statistics. Prob. Eng. Inf. Sci. 25, 449–467.Google Scholar
Burkschat, M. and Navarro, J. (2013). Dynamic signature of coherent systems based on sequential order statistics. J. Appl. Prob. 50, 272–287.Google Scholar
Casella, G. and Berger, R. L. (2002). Statistical Inference, 2nd edn. Thomson, Pacific Grove, CA.Google Scholar
Cramer, E. (2006). Sequential order statistics. In Encyclopedia of Statistical Sciences, Vol. 12, John Wiley, Hoboken, NJ, pp. 7629–7634.Google Scholar
Cramer, E. and Kamps, U. (2001). Sequential k-out-of-n systems. In Advances in Reliability (Handbook Statist. 20), North-Holland, Amesterdam, pp. 301–372.Google Scholar
David, H. A. and Nagaraja, H. N. (2003). Order Statistics, 3rd edn. John Wiley, Hoboken, NJ.Google Scholar
Gurler, S. (2012). On residual lifetimes in sequential (n-k+1)-out-of-n systems. Statist. Papers 53, 23–31.Google Scholar
Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994). Continuous Univariate Distributions, Vol. 1, 2nd edn. John Wiley, New York.Google Scholar
Kamps, U. (1995). A Concept of Generalized Order Statistics. Teubner, Stuttgart.Google Scholar
Kvam, P. H. and PeΓ±a, E. A. (2005). Estimating load-sharing properties in a dynamic reliability system. J. Amer. Statist. Assoc. 100, 262–272.Google Scholar
Lai, C.-D. and Xie, M. (2006). Stochastic Ageing and Dependence for Reliability. Springer, New York.Google Scholar
Marshall, A. W. and Olkin, I. (2007). Life Distributions. Springer, New York.Google Scholar
MΓΌller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, Chichester.Google Scholar
Murthy, D. N. P., Xie, M. and Jiang, R. (2004). Weibull Models. John Wiley, Hoboken, NJ.Google Scholar
Navarro, J. and Burkschat, M. (2011). Coherent systems based on sequential order statistics. Naval Res. Logistics 58, 123–135.Google Scholar
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.Google Scholar
Torrado, N., Lillo, R. E. and Wiper, M. P. (2012). Sequential order statistics: ageing and stochastic orderings. Methodol. Comput. Appl. Prob. 14, 579–596.Google Scholar