Skip to main content
SpringerLink
Log in

Multivariate Comparisons of Ordered Data

  • Chapter
  • First Online:
Stochastic Orders in Reliability and Risk

Part of the book series: Lecture Notes in Statistics ((LNSP,volume 208))

Abstract

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

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Bibliography

  1. Asadi, M.: On the mean past lifetime of the components of a parallel system. Journal of Statistical Planning and Inference, 136, 1197–1206 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. 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. Avérous, J., Genest, C., and Kochar, S. C.: On the dependence structure of order statistics. Journal of Multivariate Analysis, 94, 159–171 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. 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)

    Article  MathSciNet  MATH  Google Scholar 

  5. 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)

    Article  MathSciNet  Google Scholar 

  6. Bartosewicz, J.: Dispersive ordering and the total time on test transform. Statistics and Probability Letters, 4, 285–288 (1986)

    Article  MathSciNet  Google Scholar 

  7. 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)

    Article  MathSciNet  MATH  Google Scholar 

  8. 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)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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)

    Article  MathSciNet  MATH  Google Scholar 

  10. 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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  12. 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)

    Article  MathSciNet  MATH  Google Scholar 

  13. 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)

    Article  MathSciNet  MATH  Google Scholar 

  14. 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. 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. Chandler, K. N.: The distribution and frequency of record values. Journal of the Royal Statistical Society, Series B, 14, 220–228 (1952)

    MathSciNet  MATH  Google Scholar 

  17. Chen, J. and Hu, T.: Multivariate dispersive ordering of generalized order statistics. Journal of Iranian Statistical Society, 6, 61–75 (2007)

    Google Scholar 

  18. 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)

    MathSciNet  MATH  Google Scholar 

  19. 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)

    Article  MATH  Google Scholar 

  20. 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)

    MathSciNet  Google Scholar 

  21. 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)

    Article  MATH  Google Scholar 

  22. 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)

    Article  MathSciNet  MATH  Google Scholar 

  23. 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)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hu, T. and Zhuang, W.: A note on stochastic comparisons of generalized order statistics. Statistics and Probability Letters, 72, 163–170 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kamps, U.: A concept of generalized order statistics. Journal of Statistical Planning and Inference, 48, 1–23 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kamps, U.: A Concept of Generalized Order Statistics. B.G. Taubner, Stuttgart (1995)

    Book  MATH  Google Scholar 

  27. 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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  29. Khaledi, B.-E. and Kochar, S. C.: Dependence orderings of generalized order statistics. Statistics and Probability Letters, 73, 357–367 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Khaledi, B.-E. and Shaked, M.: Ordering conditional lifetimes of coherent systems. Journal of Statistical Planning and Inference, 137, 1173–1184 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. Khaledi, B.-E. and Shojaei, R.: On stochastic orderings between residual record values. Statistics and Probability Letters, 77, 1467–1472 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kochar, S. C.: On stochastic ordering between distributions and their sample spacings. Statistics and Probability Letters, 42, 345–352 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kochar, S. C. and Xu, M.: A new dependence ordering with applications. Journal of Multivariate Analysis, 99, 2172–2184 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. 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)

    Article  MathSciNet  MATH  Google Scholar 

  35. Langberg, N., Leon, R. V. and Proschan, F.: Characterization of nonparametric classes of life distributions. Annals of Probability, 8, 1163–1170 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  36. 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)

    Article  MathSciNet  MATH  Google Scholar 

  37. 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)

    Article  Google Scholar 

  38. 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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  40. Marinacci, M. and Montrucchio, L.: Ultramodular functions. Mathematics of Operations Research, 30, 311–332 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  41. Müller, A. and Stoyan, D.: Comparison Methods for Stochastic Models and Risks. Wiley, Chichester (2002)

    MATH  Google Scholar 

  42. Navarro, J.: Likelihood ratio ordering of order statistics, mixtures and systems. Journal of Statistical Planning and Inference, 138, 1242–1257 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  43. 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. Navarro, J. and Shaked, M.: Hazard rate ordering of order statistics and systems. Journal of Applied Probability, 43, 391–408 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  45. Nelsen, R. B.: An Introduction to Copulas (2nd Edition). Springer (2006)

    Google Scholar 

  46. Pellerey, F., Shaked, M. and Zinn, J.: Nonhomogeneous Poisson processes and logconcavity. Probability in the Engineering and Informational Sciences, 14, 353–373 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  47. Pfeifer, D.: ’Record values’ in einem stochastischen Modell mit nicht-identischen Verteilungen. Dissertation, Aachen University of Technology (1979)

    Google Scholar 

  48. Qiu, G. and Wu, J.: Some comparisons between generalized order statistics. Applied Mathematics: A Journal of Chinese Universities, Series B, 22, 325–333 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  49. 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. Shaked, M. and Shanthikumar, J. G.: Multivariate imperfect repair. Operations Research, 34, 437–448 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  51. Shaked, M. and Shanthikumar, J. G.: Stochastic Orders and Their Applications. Academic Press, San Diego (1994)

    MATH  Google Scholar 

  52. Shaked, M. and Shanthikumar, J. G.: Two variability orders. Probability in the Engineering and Informational Sciences, 12, 1–23 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  53. Shaked, M. and Shanthikumar, J. G.: Stochastic Orders. Springer, New York (2007)

    Book  MATH  Google Scholar 

  54. Shaked, M. and Szekli, R.: Comparison of replacement policies via point processes. Advances in Applied Probability, 27, 1079–1103 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  55. 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)

    MathSciNet  MATH  Google Scholar 

  56. Xie, H. and Hu, T.: Conditional ordering of generalized order statistics. Probability in the Engineering and Informational Sciences, 22, 333–346 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  57. Xie, H. and Hu, T.: Ordering p-spacings of generalized order statistics revisited. Probability in the Engineering and Informational Sciences, 23, 1–16 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  58. Xie, H. and Hu, T.: Some new results on multivariate dispersive ordering of generalized order statistics. Journal of Multivariate Analysis, 101, 964–970 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  59. Zhao, P. and Balakrishnan, N.: Stochastic comparisons of conditional generalized order statistics. Journal of Statistical Planning and Inference, 139, 2920–2932 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  60. 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)

    Article  MathSciNet  MATH  Google Scholar 

  61. Zhuang, W. and Hu, T.: Multivariate stochastic comparisons of sequential order statistics. Probability in the Engineering and Informational Sciences, 21, 47–66 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  62. Zhuang, W., Yao, J. and Hu, T.: Conditional ordering of order statistics. Journal of Multivariate Analysis, 101, 640–644 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departmento Estadística e Investigación Operativa, Universidad de Murcia, 30100, Espinardo, Murcia, Spain

    Félix Belzunce

Authors
  1. Félix Belzunce
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Félix Belzunce .

Editor information

Editors and Affiliations

  1. Dept. Mathematics, Washington State University, Lighty 370, Pullman, 99164, Washington, USA

    Haijun Li

  2. Xiamen University, Xiamen, 361000, China, People's Republic

    Xiaohu Li

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this chapter

Cite this chapter

Belzunce, F. (2013). Multivariate Comparisons of Ordered Data. In: Li, H., Li, X. (eds) Stochastic Orders in Reliability and Risk. Lecture Notes in Statistics(), vol 208. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6892-9_4

Download citation

Publish with us

Policies and ethics

Search

Navigation