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Multivariate Comparisons of Ordered Data

  • Félix Belzunce
Chapter
Part of the Lecture Notes in Statistics book series (LNS, volume 208)

Abstract

In this paper we present a review for some of the main results about multivariate comparisons of ordered data.

Bibliography

  1. [17]
    Asadi, M.: On the mean past lifetime of the components of a parallel system. Journal of Statistical Planning and Inference, 136, 1197–1206 (2006)MathSciNetMATHCrossRefGoogle Scholar
  2. [18]
    Asadi, M. and Bairamov, I.: A note on the mean residual life function of a parallel system. Communications in Statistics – Theory and Methods, 34, 475–484 (2005)Google Scholar
  3. [23]
    Avérous, J., Genest, C., and Kochar, S. C.: On the dependence structure of order statistics. Journal of Multivariate Analysis, 94, 159–171 (2005)MathSciNetMATHCrossRefGoogle Scholar
  4. [27]
    Balakrishnan, N., Belzunce, F., Hami, N. and Khaledi, B.-E.: Univariate and multivariate likelihood ratio ordering of generalized order statistics and associated conditional variables. Probability in the Engineering and Informational Sciences, 24, 441–455 (2010)MathSciNetMATHCrossRefGoogle Scholar
  5. [28]
    Balakrishnana, N., Belzunce, F., Sordo, M. A. and Suárez-Llorensc, A.: Increasing directionally convex orderings of random vectors having the same copula, and their use in comparing ordered data. Journal of Multivariate Analysis, 105, 45–54 (2012)MathSciNetCrossRefGoogle Scholar
  6. [42]
    Bartosewicz, J.: Dispersive ordering and the total time on test transform. Statistics and Probability Letters, 4, 285–288 (1986)MathSciNetCrossRefGoogle Scholar
  7. [48]
    Belzunce, F., Franco, M. and Ruiz, J. M.: On aging properties based on the residual life of k-out-of-n systems. Probability in the Engineering and Informational Sciences, 13, 193–199 (1999)MathSciNetMATHCrossRefGoogle Scholar
  8. [50]
    Belzunce, F., Gurler, S. and Ruiz, J. M.: Revisiting multivariate likelihood ratio ordering results for order statistics. Probability in the Engineering and Informational Sciences, 25, 355–368 (2011)MathSciNetMATHCrossRefGoogle Scholar
  9. [51]
    Belzunce, F., Lillo, R. E., Ruiz, J. M. and Shaked, M.: Stochastic comparisons of nonhomogeneous processes. Probability in the Engineering and Informational Sciences, 15, 199–224 (2001)MathSciNetMATHCrossRefGoogle Scholar
  10. [52]
    Belzunce, F., Lillo, R., Ruiz, J. M. and Shaked, M.: Stochastic ordering of record and inter-record values. Recent developments in ordered random variables, 119–137, Nova Science Publishers, New York (2007)Google Scholar
  11. [54]
    Belzunce, F., Mercader, J. A. and Ruiz, J. M.: Stochastic comparisons of generalized order statistics. Probability in the Engineering and Informational Sciences, 19, 99–120 (2005)MathSciNetMATHCrossRefGoogle Scholar
  12. [56]
    Belzunce, F., Ruiz, J. M. and Suárez-Llorens, A.: On multivariate dispersion orderings based on the standard construction. Statistics and Probability Letters, 78, 271–281 (2008)MathSciNetMATHCrossRefGoogle Scholar
  13. [57]
    Belzunce, F. and Shaked, M.: Stochastic comparisons of mixtures of convexly ordered distributions with applications in reliability theory. Statistics and Probability Letters, 53, 363–372 (2001)MathSciNetMATHCrossRefGoogle Scholar
  14. [73]
    Boland, P. J., Hu, T., Shaked, M. and Shanthikumar, J. G.: Stochastic ordering of order statistics II. Modelling Uncertainty: An Examination of Stochastic Theory, Methods and Applications (Eds. M. Dror, P. L’Ecuyer and F. Szidarovszky, Kluwer, Boston), 607–623 (2002)Google Scholar
  15. [74]
    Boland, P. J., Shaked, M. and Shanthikumar, J. G.: Stochastic ordering of order statistics. Handbook of Statistics (Eds. N. Balakrishnan and C.R. Rao), 16, 89–103 (1998)Google Scholar
  16. [83]
    Chandler, K. N.: The distribution and frequency of record values. Journal of the Royal Statistical Society, Series B, 14, 220–228 (1952)MathSciNetMATHGoogle Scholar
  17. [86]
    Chen, J. and Hu, T.: Multivariate dispersive ordering of generalized order statistics. Journal of Iranian Statistical Society, 6, 61–75 (2007)Google Scholar
  18. [156]
    Fang, Z., Hu, T., Wu, Y. and Zhuang, W.: Multivariate stochastic orderings of spacings of generalized order statistics. Chinese Journal fo Applied Probability and Statistics, 22, 295–303 (2006)MathSciNetMATHGoogle Scholar
  19. [161]
    Fernandez-Ponce, J. M. and Suárez-Llorens, A.: A multivariate dispersion ordering based on quantiles more widely separated. Journal of Multivariate Analisys, 85, 40–53 (2003)MATHCrossRefGoogle Scholar
  20. [168]
    Franco, M., Ruiz, J. M. and Ruiz, M. C.: Stochastic orderings between spacings of generalized order statistics. Probability in the Engineering and Informational Sciences, 16, 471–484 (2001)MathSciNetGoogle Scholar
  21. [185]
    Gupta, R. C. and Kirmani, S. N. U. A.: Closure and monotonicity properties of nonhomogeneous Poisson processes and record values. Probability in the Engineering and Informational Sciences, 2, 475–484 (1988)MATHCrossRefGoogle Scholar
  22. [198]
    Hu, T., Jin, W. and Khaledi, B.-E.: Ordering conditional distributions of generalized order statistics. Probability in the Engineering and Informational Sciences, 21, 401–417 (2007)MathSciNetMATHCrossRefGoogle Scholar
  23. [204]
    Hu, T. and Zhuang, W.: Stochastic properties of p-spacings of generalized order statistics. Probability in the Engineering and Informational Sciences, 19, 257–276 (2005)MathSciNetMATHCrossRefGoogle Scholar
  24. [205]
    Hu, T. and Zhuang, W.: A note on stochastic comparisons of generalized order statistics. Statistics and Probability Letters, 72, 163–170 (2005)MathSciNetMATHCrossRefGoogle Scholar
  25. [221]
    Kamps, U.: A concept of generalized order statistics. Journal of Statistical Planning and Inference, 48, 1–23 (1995)MathSciNetMATHCrossRefGoogle Scholar
  26. [222]
    Kamps, U.: A Concept of Generalized Order Statistics. B.G. Taubner, Stuttgart (1995)MATHCrossRefGoogle Scholar
  27. [227]
    Khaledi, B.-E.: Some new results on stochastic comparisons between generalized order statistics. Journal of the Iranian Statistical Society, 4, 35–49 (2005)Google Scholar
  28. [228]
    Khaledi, B.-E., Amiripour, F., Hu, T. and Shojaei, R.: Some new results on stochastic comparisons of record values. Communications in Statistics: Theory and Methods, 38, 2056–2066 (2009)MathSciNetMATHCrossRefGoogle Scholar
  29. [234]
    Khaledi, B.-E. and Kochar, S. C.: Dependence orderings of generalized order statistics. Statistics and Probability Letters, 73, 357–367 (2005)MathSciNetMATHCrossRefGoogle Scholar
  30. [235]
    Khaledi, B.-E. and Shaked, M.: Ordering conditional lifetimes of coherent systems. Journal of Statistical Planning and Inference, 137, 1173–1184 (2007)MathSciNetMATHCrossRefGoogle Scholar
  31. [236]
    Khaledi, B.-E. and Shojaei, R.: On stochastic orderings between residual record values. Statistics and Probability Letters, 77, 1467–1472 (2007)MathSciNetMATHCrossRefGoogle Scholar
  32. [247]
    Kochar, S. C.: On stochastic ordering between distributions and their sample spacings. Statistics and Probability Letters, 42, 345–352 (1999)MathSciNetMATHCrossRefGoogle Scholar
  33. [257]
    Kochar, S. C. and Xu, M.: A new dependence ordering with applications. Journal of Multivariate Analysis, 99, 2172–2184 (2008)MathSciNetMATHCrossRefGoogle Scholar
  34. [260]
    Kochar, S. C. and Xu, M.: On residual lifetime of k-out-of-n systems with nonidentical components. Probability in the Engineering and Informational Sciences, 24, 109–127 (2010)MathSciNetMATHCrossRefGoogle Scholar
  35. [270]
    Langberg, N., Leon, R. V. and Proschan, F.: Characterization of nonparametric classes of life distributions. Annals of Probability, 8, 1163–1170 (1980)MathSciNetMATHCrossRefGoogle Scholar
  36. [286]
    Li, X. and Chen, J.: Aging properties of the residual life of k-out-of-n systems with independent but non-identical components. Applied Stochastic Models in Business and Industry, 20, 143–153 (2004)MathSciNetMATHCrossRefGoogle Scholar
  37. [296]
    Li, X. and Zhao, P.: Some aging properties of the residual life of k-out-of-n systems. IEEE Transactions on Reliability, 55, 535–541 (2006)CrossRefGoogle Scholar
  38. [297]
    Li, X. and Zhao, P.: Stochastic comparison on general inactivity time and general residual life of k-out-of-n systems. Communications in Statistics – Simulation and Computation, 37, 1005–1019 (2008)Google Scholar
  39. [298]
    Li, X. and Zuo, M. J.: On the behaviour of some new aging properties based upon the residual life of k-out-of-n systems. Journal of Applied Probability, 39, 426–433 (2002)MathSciNetMATHCrossRefGoogle Scholar
  40. [305]
    Marinacci, M. and Montrucchio, L.: Ultramodular functions. Mathematics of Operations Research, 30, 311–332 (2005)MathSciNetMATHCrossRefGoogle Scholar
  41. [335]
    Müller, A. and Stoyan, D.: Comparison Methods for Stochastic Models and Risks. Wiley, Chichester (2002)MATHGoogle Scholar
  42. [338]
    Navarro, J.: Likelihood ratio ordering of order statistics, mixtures and systems. Journal of Statistical Planning and Inference, 138, 1242–1257 (2008)MathSciNetMATHCrossRefGoogle Scholar
  43. [347]
    Navarro, J., Rychlik, T. and Shaked, M.: Are the order statistics ordered? A survey of recent results. Communications in Statistics - Theory and Methods, 36, 1273C1290 (2007)Google Scholar
  44. [352]
    Navarro, J. and Shaked, M.: Hazard rate ordering of order statistics and systems. Journal of Applied Probability, 43, 391–408 (2006)MathSciNetMATHCrossRefGoogle Scholar
  45. [355]
    Nelsen, R. B.: An Introduction to Copulas (2nd Edition). Springer (2006)Google Scholar
  46. [372]
    Pellerey, F., Shaked, M. and Zinn, J.: Nonhomogeneous Poisson processes and logconcavity. Probability in the Engineering and Informational Sciences, 14, 353–373 (2000)MathSciNetMATHCrossRefGoogle Scholar
  47. [373]
    Pfeifer, D.: ’Record values’ in einem stochastischen Modell mit nicht-identischen Verteilungen. Dissertation, Aachen University of Technology (1979)Google Scholar
  48. [379]
    Qiu, G. and Wu, J.: Some comparisons between generalized order statistics. Applied Mathematics: A Journal of Chinese Universities, Series B, 22, 325–333 (2007)MathSciNetMATHCrossRefGoogle Scholar
  49. [406]
    Sadegh, M. K.: Mean past and mean residual life functions of a parallel system with nonidentical components. Communications in Statistics – Theory and Methods, 37, 1134–1145 (2008)Google Scholar
  50. [418]
    Shaked, M. and Shanthikumar, J. G.: Multivariate imperfect repair. Operations Research, 34, 437–448 (1986)MathSciNetMATHCrossRefGoogle Scholar
  51. [422]
    Shaked, M. and Shanthikumar, J. G.: Stochastic Orders and Their Applications. Academic Press, San Diego (1994)MATHGoogle Scholar
  52. [425]
    Shaked, M. and Shanthikumar, J. G.: Two variability orders. Probability in the Engineering and Informational Sciences, 12, 1–23 (1998)MathSciNetMATHCrossRefGoogle Scholar
  53. [426]
    Shaked, M. and Shanthikumar, J. G.: Stochastic Orders. Springer, New York (2007)MATHCrossRefGoogle Scholar
  54. [430]
    Shaked, M. and Szekli, R.: Comparison of replacement policies via point processes. Advances in Applied Probability, 27, 1079–1103 (1995)MathSciNetMATHCrossRefGoogle Scholar
  55. [461]
    Vaughan, R. J. and Venables, W. N.: Permanent expressions for order statistics densities. Journal of the Royal Statistical Society: Series B, 34, 308–310 (1972)MathSciNetMATHGoogle Scholar
  56. [478]
    Xie, H. and Hu, T.: Conditional ordering of generalized order statistics. Probability in the Engineering and Informational Sciences, 22, 333–346 (2008)MathSciNetMATHCrossRefGoogle Scholar
  57. [479]
    Xie, H. and Hu, T.: Ordering p-spacings of generalized order statistics revisited. Probability in the Engineering and Informational Sciences, 23, 1–16 (2009)MathSciNetMATHCrossRefGoogle Scholar
  58. [480]
    Xie, H. and Hu, T.: Some new results on multivariate dispersive ordering of generalized order statistics. Journal of Multivariate Analysis, 101, 964–970 (2010)MathSciNetMATHCrossRefGoogle Scholar
  59. [489]
    Zhao, P. and Balakrishnan, N.: Stochastic comparisons of conditional generalized order statistics. Journal of Statistical Planning and Inference, 139, 2920–2932 (2009)MathSciNetMATHCrossRefGoogle Scholar
  60. [491]
    Zhao, P., Li, X. and Balakrishnan, N.: Conditional ordering of k-out-of-n systems with independent but nonidentical components. Journal of Applied Probability, 45, 1113–1125 (2008)MathSciNetMATHCrossRefGoogle Scholar
  61. [493]
    Zhuang, W. and Hu, T.: Multivariate stochastic comparisons of sequential order statistics. Probability in the Engineering and Informational Sciences, 21, 47–66 (2007)MathSciNetMATHCrossRefGoogle Scholar
  62. [495]
    Zhuang, W., Yao, J. and Hu, T.: Conditional ordering of order statistics. Journal of Multivariate Analysis, 101, 640–644 (2010)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Departmento Estadística e Investigación OperativaUniversidad de MurciaEspinardo, MurciaSpain

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