Hostname: page-component-89b8bd64d-sd5qd Total loading time: 0 Render date: 2026-05-07T07:16:37.827Z Has data issue: false hasContentIssue false
  • English
  • Français

AGING PROPERTIES OF SEQUENTIAL ORDER STATISTICS

Published online by Cambridge University Press:  21 July 2011

M. Burkschat  and
J. Navarro
M. Burkschat
Affiliation:
Otto von Guericke University Magdeburg, Institute of Mathematical Stochastics, D-39016 Magdeburg, Germany E-mail: marco.burkschat@ovgu.de
J. Navarro
Affiliation:
Facultad de Matemáticas, Universidad de Murcia, 30100 Murcia, Spain E-mail: jorgenav@um.es
Get access Rights & Permissions [Opens in a new window]

Abstract

Sequential order statistics describe the ordered failure times in a k-out-of-n system, where the failures of components might affect the performance of remaining working components. In this article aging properties of sequential order statistics are examined and conditions are given such that the distribution of a sequential order statistic is ILR, IFR, IFRA, or NBU. Moreover, conditions for aging properties of spacings of sequential order statistics are obtained.

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 25 , Issue 4 , October 2011 , pp. 449 - 467
Copyright
Copyright © Cambridge University Press 2011

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

1.An, M.Y. (1998). Logconcavity versus logconvexity: a complete characterization. Journal of Economic Theory 80: 350–369.CrossRefGoogle Scholar
2.Balakrishnan, N., Beutner, E., & Kamps, U. (2008). Order restricted inference for sequential k-out-of-n systems. Journal of Multivariate Analysis 99: 1489–1502.CrossRefGoogle Scholar
3.Barlow, R.E. & Proschan, F. (1975). Statistical theory of reliability and life testing. Silver Spring, MD: To Begin With.Google Scholar
4.Belzunce, F., Mercader, J.A., & Ruiz, J.M. (2003). Multivariate aging properties of epoch times of nonhomogeneous processes. Journal of Multivariate Analysis 84: 335–350.Google Scholar
5.Beutner, E. (2008). Nonparametric inference for sequential k-out-of-n systems. Annals of the Institute of Statistical Mathematics 60: 605–626.Google Scholar
6.Block, H.W., Li, Y., & Savits, T.H. (2003). Preservation of properties under mixture. Probability in the Engineering and Informational Sciences 17: 205–212.CrossRefGoogle Scholar
7.Block, H.W. & Savits, T.H. (1980). Multivariate increasing failure rate average distributions. Annals of Probability 8: 793–801.CrossRefGoogle Scholar
8.Chen, H., Xie, H., & Hu, T. (2009). Log-concavity of generalized order statistics. Statistics and Probability Letters 79: 396–399.Google Scholar
9.Cramer, E. (2004). Logconcavity and unimodality of progressively censored order statistics. Journal of Statististical Planning and Inference 68: 83–90.Google Scholar
10.Cramer, E. (2006). Sequential order statistics. In Kotz, S., Balakrishnan, N., Read, C.B., Vidakovic, B., & Johnson, N.L. (eds.), Encyclopedia of statistical sciences, Vol. 12; 2nd ed.Hoboken, NJ: Wiley, pp. 7629–7634.Google Scholar
11.Cramer, E. & Kamps, U. (1996). Sequential order statistics and k-out-of-n systems with sequentially adjusted failure rates. Annals of the Institute of Statistical Mathematics 48: 535–549.Google Scholar
12.Cramer, E. & Kamps, U. (2001). Sequential k-out-of-n systems. In Balakrishnan, N. & Rao, C.R. (eds.), Handbook of statistics; Advances in reliability, Vol. 20. Amsterdam: Elsevier, pp. 301–372.Google Scholar
13.Cramer, E. & Kamps, U. (2003). Marginal distributions of sequential and generalized order statistics. Metrika 58: 293–310.Google Scholar
14.Esary, J.D. & Proschan, F. (1963). Relationship between system failure rate and component failure rates. Technometrics 5: 183–189.Google Scholar
15.Kalashnikov, V.V. & Rachev, S.T. (1986). Characterization of queueing models and their stability. In Prohorov, Y. K. (ed.), Probability theory and mathematical statistics, Vol. 2. Amsterdam: VNU Science Press, pp. 37–53.Google Scholar
16.Kamps, U. (1995). A concept of generalized order statistics. Stuttgart: Teubner.Google Scholar
17.Kamps, U. & Cramer, E. (2001). On distributions of generalized order statistics. Statistics 35: 269–280.CrossRefGoogle Scholar
18.Lai, C.-D. & Xie, M. (2006). Stochastic aging and dependence for reliability. New York: Springer.Google Scholar
19.Marshall, A. W. & Olkin, I. (2007). Life distributions. New York: Springer.Google Scholar
20.Navarro, J. (2008). Likelihood ratio ordering of order statistics, mixtures and systems. Journal of Statististical Planning and Inference 138: 1242–1257.CrossRefGoogle Scholar
21.Navarro, J. & Burkschat, M. (2011). Coherent systems based on sequential order statistics. Naval Research Logistics 58: 123–135.Google Scholar
22.Navarro, J. & Shaked, M. (2010). Some properties of the minimum and the maximum of random variables with joint logconcave distributions. Metrika 71: 313–317.CrossRefGoogle Scholar
23.Pellerey, F., Shaked, M., & Zinn, J. (2000). Nonhomogeneous Poisson processes and logconcavity. Probability in the Engineering and Informational Sciences 14: 353–373.CrossRefGoogle Scholar
24.Prékopa, A. (1973). On logarithmic concave measures and functions. Acta Scientiarum Mathematicarum 34: 335–343.Google Scholar
25.Righter, R., Shaked, M., & Shanthikumar, J.G. (2009). Intrinsic aging and classes of nonparametric distributions. Probability in the Engineering and Informational Sciences 23: 563–582.Google Scholar
26.Rowell, G. & Siegrist, K. (1998). Relative aging of distributions. Probability in the Engineering and Informational Sciences 12: 469–478.Google Scholar
27.Sengupta, D. & Deshpande, J.V. (1994). Some results on the relative ageing of two life distributions. Journal of Applied Probability 31: 991–1003.CrossRefGoogle Scholar
28.Shaked, M. & Shantikumar, J.G. (1991). Dynamic multivariate aging notions in reliability theory. Stochastic Processes and their Applications 38: 85–97.Google Scholar
29.Shanthikumar, J.G. & Baxter, L.A. (1985). Closure properties of the relevation transform. Naval Research Logistics 32: 185–189.Google Scholar
30.Takahasi, K. (1988). A note on hazard rates of order statistics. Communications in Statistics: Theory and Methods 17: 4133–4136.Google Scholar
31.Zhuang, W. & Hu, T. (2007). Multivariate stochastic comparisons of sequential order statistics. Probability in the Engineering and Informational Sciences 21: 47–66.CrossRefGoogle Scholar