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Order statistics with memory: a model with reliability applications

Part of: Nonparametric inference Distribution theory - Probability

Published online by Cambridge University Press:  09 December 2016

Alexander Katzur  and
Udo Kamps
Alexander Katzur*
Affiliation:
RWTH Aachen University
Udo Kamps*
Affiliation:
RWTH Aachen University
*
* Postal address: Institute of Statistics, RWTH Aachen University, 52056 Aachen, Germany.
* Postal address: Institute of Statistics, RWTH Aachen University, 52056 Aachen, Germany.
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Abstract

An extended model of order statistics based on possibly different distributions is introduced and analyzed. In the interpretation of successive failure times in a 𝑘-out-of-𝑛 system, say, until each failure, the time periods under previous (increasing) loads exerted on the remaining components are recorded. Then the lifetime distribution of the system depends on the complete failure scheme. Thus, order statistics with memory provide an alternative to the use of sequential order statistics, which form a Markov chain. The quantities as well as their spacings, the interoccurrence times, can be compared by means of stochastic ordering.

Keywords

Order statistics𝑘-out-of-𝑛 systemsequential order statisticsordered data

MSC classification

Primary: 60E05: Distributions
Secondary: 60E15: Inequalities; stochastic orderings 62G30: Order statistics; empirical distribution functions

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 4 , December 2016 , pp. 974 - 988
Copyright
Copyright © Applied Probability Trust 2016 

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References

Bedbur, S.,Beutner, E. and Kamps, U. (2012).Generalized order statistics: an exponential family in model parameters.Statistics 46,159–166.Google Scholar
Bedbur, S.,Kamps, U. and Kateri, M. (2015).Meta-analysis of general step-stress experiments under repeated Type-II censoring.Appl. Math. Modelling 39,2261–2275.Google Scholar
Beutner, E. (2010).Nonparametric model checking for k-out-of-n systems.J. Statist. Planning Inference 140,626–639.Google Scholar
Burkschat, M. (2009).Systems with failure-dependent lifetimes of components.J. Appl. Prob. 46,1052–1072.Google 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
Cramer, E. and Kamps, U. (2001).Sequential k-out-of-n systems.In Advances in Reliability(Handbook Statist. 20),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.Google Scholar
David, H. A. and Nagaraja, H. N. (2003).Order Statistics,3rd edn.John Wiley,Hoboken, NJ.Google Scholar
Deshpande, J. V.,Dewan, I. and Naik-Nimbalkar, U. V. (2010).A family of distributions to model load sharing systems.J. Statist. Planning Inference 140,1441–1451.Google Scholar
Kamps, U. (1995).A concept of generalized order statistics.J. Statist. Planning Inference 48,1–23.Google Scholar
Khaledi, B.-E. and Kochar, S. (2005).Dependence orderings for generalized order statistics.Statist. Prob. Lett. 73,357–367.Google Scholar
Shaked, M. and Shanthikumar, J. G. (2007).Stochastic Orders.Springer,New York.Google Scholar
Unnikrishnan Nair, N. and Vineshkumar, B. (2011).Ageing concepts: an approach based on quantile function.Statist. Prob. Lett. 81,2016–2025.Google Scholar
Unnikrishnan Nair, N.,Sankaran, P. G. and Balakrishnan, N. (2013).Quantile-Based Reliability Analysis.Birkhäuser,New York.Google Scholar
Vuong, Q. N.,Bedbur, S. and Kamps, U. (2013).Distances between models of generalized order statistics.J. Multivariate Anal. 118,24–36.Google Scholar
Xie, H. and Zhuang, W. (2011).Some new results on ordering of simple spacings of generalized order statistics.Prob. Eng. Inf. Sci. 25,71–81.Google Scholar