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Stochastic Ordering of Order Statistics II

  • Philip J. Boland
  • Taizhong Hu
  • Moshe Shaked
  • J. George Shanthikumar
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 46)

Abstract

In this paper we survey some recent developments involving comparisons of order statistics and spacings in various stochastic senses.

Keywords

Reliability theory k-out-of-n systems IFR DFR hazard rate order likelihood ratio order dispersive order sample spacings 

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References

  1. Alzaid, A. A. and F. Proschan. (1992). Dispersivity and stochastic majorization. Statistics and Probability Letters13, 275–278.MathSciNetCrossRefGoogle Scholar
  2. Arnold, B. C. and J. A. Villasenor. (1998). Lorenz ordering of order statistics and record values. In Handbook of Statistics, Volume 16 (Eds): N. Balakrishnan and C. R. Rao), Elsevier, Amsterdam, 75–87.zbMATHGoogle Scholar
  3. Bagai, I. and S.C. Kochar. (1986). On tail-ordering and comparison of failure rates. Communications in Statistics-Theory and Methods15, 1377–1388.MathSciNetCrossRefGoogle Scholar
  4. Barlow, R. E. and F. Proschan. (1966). Inequalities for linear combinations of order statistics from restricted families. Annals of Mathematical Statistics37, 1574–1592.MathSciNetCrossRefGoogle Scholar
  5. Barlow, R. E. and F. Proschan. (1975). Statistical Theory of Reliability and Life Testing, Probability Models, Holt, Rinehart, and Winston, New York, NY.zbMATHGoogle Scholar
  6. Bartoszewicz, J. (1985). Dispersive ordering and monotone failure rate distributions. Advances in Applied Probability17, 472–474.MathSciNetCrossRefGoogle Scholar
  7. Bartoszewicz, J. (1986). Dispersive ordering and the total time on test transformation. Statistics and Probability Letters4, 285–288.MathSciNetCrossRefGoogle Scholar
  8. Bartoszewicz, J. (1998a). Applications of a general composition theorem to the star order of distributions. Statistics and Probability Letters38, 1–9.MathSciNetCrossRefGoogle Scholar
  9. Bartoszewicz, J. (1998b). Characterizations of the dispersive order of distributions by the Laplace transform. Statistics and Probability Letters40, 23–29.MathSciNetCrossRefGoogle Scholar
  10. Belzunce, F., M. Franco, J.-M. Ruiz, and M. C. Ruiz. (2001). On partial orderings between coherent systems with different structures. Probability in the Engineering and Informational Sciences15, 273–293.MathSciNetCrossRefGoogle Scholar
  11. Block, H. W., T.H. Savits, and H. Singh. (1998). The reversed hazard rate function. Probability in the Engineering and Informational Sciences12, 69–90.MathSciNetCrossRefGoogle Scholar
  12. Boland, P. J., E. El-Neweihi, and F. Proschan. (1994). Applications of the hazard rate ordering in reliability and order statistics. Journal of Applied Probability31, 180–192.MathSciNetCrossRefGoogle Scholar
  13. Boland, P. J. and F. Proschan. (1994). Stochastic order in system reliability theory. iIn Stochastic Orders and Their Applications (Eds: M. Shaked and J. G. Shanthikumar), Academic Press, San Diego, 485–508.Google Scholar
  14. Boland, P. J., M. Shaked, and J.G. Shanthikumar. (1998). Stochastic ordering of order statistics. In Handbook of Statistics, Volume 19 (Eds): N. Balakrishnan and C. R. Rao), Elsevier, Amsterdam, 89–103.zbMATHGoogle Scholar
  15. Dykstra, R., S. Kochar, and J. Rojo. (1997). Stochastic comparisons of parallel systems of heterogeneous exponential components. Journal of Statistical Planning and Inference65, 203–211.MathSciNetCrossRefGoogle Scholar
  16. Hu, T. and F. He. (2000). A note on comparisons of k-out-of-n systems with respect to the hazard and reversed hazard rate orders. Probability in the Engineering and Informational Sciences14, 27–32.MathSciNetCrossRefGoogle Scholar
  17. Hu, T. and Y. Wei. (2000). Stochastic comparisons of spacings from restricted families of distributions. Technical Report, Department of Statistics and Finance, University of Science and Technology of China.Google Scholar
  18. Joag-Dev, K. (1995). Personal communication.Google Scholar
  19. Kamps, O. (1995). A Concept of Generalized Order Statistics, B. G. Taubner, Stuttgart.CrossRefGoogle Scholar
  20. Khaledi, B. and S. Kochar. (1999). Stochastic orderings between distributions and their sample spacings-II. Statistics and Probability Letters44, 161–166.MathSciNetCrossRefGoogle Scholar
  21. Khaledi, B. and S. Kochar. (2000a). On dispersive ordering between order statistics in one-sample and two-sample problems. Statistics and Probability Letters46, 257–261.MathSciNetCrossRefGoogle Scholar
  22. Khaledi, B. and S. Kochar. (2000b). Stochastic properties of spacings in a single outlier exponential model. Technical Report, Indian Statistical Institute.Google Scholar
  23. Khaledi, B. and S. Kochar. (2000c). Some new results on stochastic comparisons of parallel systems. Technical Report, Indian Statistical Institute.MathSciNetCrossRefGoogle Scholar
  24. Kochar, S. C. (1996). Dispersive ordering of order statistics. Statistics and Probability Letters27, 271–274.MathSciNetCrossRefGoogle Scholar
  25. Kochar, S. C. (1998). Stochastic comparisons of spacings and order statistics. In Frontiers in Reliability (Eds: A. P. Basu, S. K. Basu and S. Mukhopadhyay), World Scientific, Singapore, 201–216.CrossRefGoogle Scholar
  26. Kochar, S. C. and S.N.U.A. Kirmani. (1995). Some results on normalized spacings from restricted families of distributions. Journal of Statistical Planning and Inference46, 47–57.MathSciNetCrossRefGoogle Scholar
  27. Kochar, S. C. and R. Korwar. (1996). Stochastic orders for spacings of heterogeneous exponential random variables. Journal of Multivariate Analysis57, 69–83.MathSciNetCrossRefGoogle Scholar
  28. Kochar, S. C. and J. Rojo. (1996). Some new results on stochastic comparisons of spacings from heterogeneous exponential distributions. Journal of Multivariate Analysis59, 272–281.MathSciNetCrossRefGoogle Scholar
  29. Li, X., Z. Li, and B-Y. Jing. (2000). Some results about the NBUC class of life distributions. Statistics and Probability Letters46, 229–237.MathSciNetCrossRefGoogle Scholar
  30. Lillo, R. E., A.K. Nanda, and M. Shaked. (2001). Preservation of some likelihood ratio stochastic orders by order statistics. Statistics and Probability Letters51, 111–119.MathSciNetCrossRefGoogle Scholar
  31. Misra, N. and E.C. van der Meulen. (2001), On stochastic properties of m-spacings, Technical Report, Department of Mathematics, Katholieke University Leuven.Google Scholar
  32. Nanda, A. K., K. Jain, and H. Singh. (1998). Preservation of some partial orderings under the formation of coherent systems. Statistics and Probability Letters39, 123–131.MathSciNetCrossRefGoogle Scholar
  33. Nanda, A. K. and M. Shaked. (2000). The hazard rate and the reversed hazard rate orders, with applications to order statistics. Annals of the Institute of Statistical Mathematics, to appear.Google Scholar
  34. Oja, H. (1981). On location, scale, skewness and kurtosis of univariate distributions. Scandinavian Journal of Statistics8, 154–168.MathSciNetzbMATHGoogle Scholar
  35. Pledger, G. and F. Proschan. (1971). Comparisons of order statistics and spacings from heterogeneous distributions. In Optimizing Methods in Statistics (Ed: J. S. Rustagi), Academic Press, New York, 89–113.zbMATHGoogle Scholar
  36. Raqab, M. Z. and W.A. Amin. (1996). Some ordering results on order statistics and record values. IAPQR Transactions21, 1–8.MathSciNetzbMATHGoogle Scholar
  37. Rojo, J. and G.Z. He. (1991). New properties and characterizations of the dispersive ordering. Statistics and Probability Letters11, 365–372.MathSciNetCrossRefGoogle Scholar
  38. Shaked, M. and J.G. Shanthikumar. (1994). Stochastic Orders and Their Applications, Academic Press, Boston.zbMATHGoogle Scholar
  39. Shanthikumar, J. G. and D.D. Yao. (1986). The preservation of likelihood ratio ordering under convolution. Stochastic Processes and Their Applications23, 259–267.MathSciNetCrossRefGoogle Scholar
  40. Wilfling, B. (1996). Lorenz ordering of power-function order statistics. Statistics and Probability Letters30, 313–319.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2002

Authors and Affiliations

  • Philip J. Boland
    • 1
  • Taizhong Hu
    • 2
  • Moshe Shaked
    • 3
  • J. George Shanthikumar
    • 4
  1. 1.Department of StatisticsUniversity College Dublin BelfieldDublin 4Ireland
  2. 2.Department of Statistics and FinanceUniversity of Science and TechnologyHefeiPeople’s Republic of China
  3. 3.Department of MathematicsUniversity of ArizonaTucsonUSA
  4. 4.Industrial Engineering & Operations ResearchUniversity of CaliforniaBerkeleyUSA

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