Some Shifted Stochastic Orders

  • Rosa E. Lillo
  • Asok K. Nanda
  • Moshe Shaked
Part of the Statistics for Industry and Technology book series (SIT)


In this paper we study some known and some new shifted orders. We compare them, we obtain some basic properties of them, we derive some closure properties of them, and we show how they can be used for stochastic comparisons of order statistics; that is, of k-out-of-n systems.

Keywords and phrases

Logconcavity logconvexity stochastic orders order statistics hazard rate likelihood ratio k-out-of-n systems 


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  1. 1.
    Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing, Probability Models, New York: Holt, Rinehart, and Winston.Google Scholar
  2. 2.
    Belzunce, F., Lillo, R., Ruiz, J.-M. and Shaked, M. (1999). Stochastic comparisons of nonhomogeneous processes. Technical Report, Department of Mathematics, University of Arizona.Google Scholar
  3. 3.
    Brown, M. and Shanthikumar, J. G. (1998), Comparing the variability of random variables and point processes, Probability in the Engineering and Informational Sciences, 12, 425–444.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Karlin, S. (1968). Total Positivity, Palo Alto: Stanford University Press.zbMATHGoogle Scholar
  5. 5.
    Müller, A. (1997). Stochastic orders generated by integrals: A unified approach, Advances in Applied Probability, 29, 414–42Google Scholar
  6. 6.
    Nakai, T. (1995). A partially observable decision problem under a shifted likelihood ratio ordering, Mathematical and Computer Modelling, 22, 237–246.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Raqab, M. Z. and Amin, W. A. (1996). Some ordering results on order statistics and record values, IAPQR Transactions, 21, 1–8.MathSciNetzbMATHGoogle Scholar
  8. 8.
    Sengupta, D. and Nanda, A. K. (1999). Log-concave and concave distributions in reliability theory, Naval Research Logistics, 46, 419–433.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Shaked, M. and Shanthikumar, J. G. (1987). Characterization of some first passage times using log-concavity and log-convexity as aging notions, Probability in the Engineering and Informational Sciences, 1, 279–291.zbMATHCrossRefGoogle Scholar
  10. 10.
    Shaked, M. and Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications, Boston: Academic Press.zbMATHGoogle Scholar
  11. 11.
    Shanthikumar, J. G. and Yao, D. D. (1986). The preservation of likelihood ratio ordering under convolutions, Stochastic Processes and Their Applications, 23, 259–267.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • Rosa E. Lillo
    • 1
  • Asok K. Nanda
    • 2
  • Moshe Shaked
    • 3
  1. 1.Universidad Carlos III de MadridMadridSpain
  2. 2.Panjab UniversityChandigarhIndia
  3. 3.University of ArizonaTucsonUSA

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