Advertisement

Sankhya A

, Volume 80, Issue 1, pp 110–120 | Cite as

Multivariate Order Statistics: the Intermediate Case

  • Michael Falk
  • Florian WisheckelEmail author
Article
  • 82 Downloads

Abstract

Asymptotic normality of intermediate order statistics taken from univariate iid random variables is well-known. We generalize this result to random vectors in arbitrary dimension, where the order statistics are taken componentwise.

Keywords and phrases

Multivariate order statistics Intermediate order statistics Copula Domain of attraction D-norm von Mises type conditions Asymptotic normality 

AMS (2000) subject classification

Primary 62G30 Secondary 62H10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arnold B. C., Balakrishnan N. and Nagaraja H. N. (2008). A First Course in Order Statistics. Society for Industrial and Applied Mathematics, Philadelphia.CrossRefzbMATHGoogle Scholar
  2. Balkema A. A. and de Haan L. (1978a). Limit distributions for order statistics i. Theory Probab. Appl. 23, 77–92.Google Scholar
  3. Balkema A. A. and de Haan L. (1978b). Limit distributions for order statistics ii. Theory Probab. Appl. 23, 341–358.Google Scholar
  4. Balkema A. A. and Resnick S. I. (1977). Max-infinite divisibility. J. Appl. Probab. 14, 309–319. doi: 10.2307/3213001.MathSciNetCrossRefzbMATHGoogle Scholar
  5. Barakat H. M. (2001). The asymptotic distribution theory of bivariate order statistics. Ann. Inst. Statist. Math. 53, 487–497.MathSciNetCrossRefzbMATHGoogle Scholar
  6. Barakat H. M. and Nigm E. M. (2012). On multivariate order statistics: Asymptotic theory and computing moments. Kuwait J. Sci. Eng. 39, 113–127.MathSciNetGoogle Scholar
  7. Barakat H. M., Nigm E. M. and Al-Awady M. A. (2015). Asymptotic properties of multivariate order statistics with random index. Bulletin of the Malaysian Mathematical Sciences Society 38, 289–301. doi: 10.1007/s40840-014-0019-7.MathSciNetCrossRefzbMATHGoogle Scholar
  8. Beirlant J., Goegebeur Y., Segers J. and Teugels J. (2004). Statistics of Extremes: Theory and Applications. Wiley Series in Probability and Statistics. Wiley, Chichester. doi: 10.1002/0470012382.CrossRefzbMATHGoogle Scholar
  9. Billingsley P. (2012). Probability and Measure. Wiley Series in Probability and Statistics, Anniversary ed. Wiley, New York.zbMATHGoogle Scholar
  10. Charpentier A. and Segers J. (2009). Tails of multivariate Archimedean copulas. J. Multivariate Anal. 100, 1521–1537. doi: 10.1016/j.jmva.2008.12.015.MathSciNetCrossRefzbMATHGoogle Scholar
  11. Cheng S., de Haan L. and Yang J. (1997). Asymptotic distributions of multivariate intermediate order statistics. Theory Probab. Appl. 41, 646–656. doi: 10.1137/S0040585X97975733.
  12. Cooil B. (1985). Limiting multivariate distributions of intermediate order statistics. Ann. Probab. 13, 469–477.MathSciNetCrossRefzbMATHGoogle Scholar
  13. David H. (1981). Order Statistics. Wiley Series in Probability and Mathematical Statics, 2nd ed. Wiley, New York.Google Scholar
  14. David H. and Nagaraja H. (2004). Order Statistics. Wiley Series in Probability and Mathematical Statics, 3rd ed. Wiley, New York.Google Scholar
  15. Falk M. (1989). A note on uniform asymptotic normality of intermediate order statistics. Ann. Inst. Statist. Math. 41, 19–29.MathSciNetzbMATHGoogle Scholar
  16. Falk M. 2016 An offspring of multivariate extreme value theory: D-norms, http://www.statistik-mathematik.uni-wuerzburg.de/fileadmin/10040800/D-norms-tutorial_book.pdf.
  17. Falk M., Hüsler J. and Reiss R.-D. (2011). Laws of Small Numbers: Extremes and Rare Events, 3rd ed. Springer, Basel. doi: 10.1007/978-3-0348-0009-9.CrossRefzbMATHGoogle Scholar
  18. Galambos J. (1975). Order statistics of samples from multivariate distributions. Journal of the American Statistical Association 70, 674–680.MathSciNetzbMATHGoogle Scholar
  19. Galambos J. (1987). The Asymptotic Theory of Extreme Order Statistics, 2nd ed. Krieger, Malabar.zbMATHGoogle Scholar
  20. de Haan L. and Ferreira A. (2006). Extreme Value Theory: An Introduction. Springer Series in Operations Research and Financial Engineering. Springer, New York. doi: 10.1007/0-387-34471-3. See http://people.few.eur.nl/ldehaan/EVTbook.correction.pdf and http://home.isa.utl.pt/~anafh/corrections.pdf for corrections and extensions.
  21. de Haan L. and Resnick S. (1977). Limit theory for multivariate sample extremes. Probab. Theory Related Fields 40, 317–337. doi: 10.1007/BF00533086.
  22. McNeil A. J. and Nešlehová J. (2009). Multivariate archimedean copulas, d-monotone functions and 1-norm symmetric distributions. Ann. Statist. 37, 3059–3097 . doi: 10.1214/07-AOS556.MathSciNetCrossRefzbMATHGoogle Scholar
  23. Nelsen R. B. (2006). An Introduction to Copulas. Springer Series in Statistics, 2nd ed. Springer, New York. doi: 10.1007/0-387-28678-0.Google Scholar
  24. Reiss R.-D. (1989). Approximate Distributions of Order Statistics: With Applications to Nonparametric Statistics. Springer Series in Statistics. Springer, New York. doi: 10.1007/978-1-4613-9620-8.CrossRefzbMATHGoogle Scholar
  25. Resnick S. I. (1987). Extreme Values, Regular Variation, and Point Processes, Applied Probability, vol. 4. Springer, New York. doi: 10.1007/978-0-387-75953-1. First Printing.CrossRefzbMATHGoogle Scholar
  26. Sklar A. (1959). Fonctions de répartition à n dimensions et leurs marges. Pub. Inst. Stat. Univ. Paris 8, 229–231.zbMATHGoogle Scholar
  27. Sklar A. (1996). Random variables, distribution functions, and copulas – a personal look backward and forward. In Distributions with fixed marginals and related topics (L. Rüschendorf, B. Schweizer, and M. D. Taylor, eds.), Lecture Notes – Monograph Series, vol. 28, 1–14. Institute of Mathematical Statistics, Hayward. doi: 10.1214/lnms/1215452606.Google Scholar
  28. Smirnov N. V. (1967). Some remarks on limit laws for order statistics. Theory. Probab. Appl. 12, 337–339.CrossRefGoogle Scholar
  29. Vatan P. (1985). Max-infinite divisibility and max-stability in infinite dimensions. In Probability in Banach Spaces V: Proceedings of the International Conference held in Medford, USA, July 16-27, 1984 (A. Beck, R. Dudley, M. Hahn, J. Kuelbs, and M. Marcus, eds.), Lecture Notes in Mathematics, vol. 1153, 400–425. Springer, Berlin. doi: 10.1007/BFb0074963.

Copyright information

© Indian Statistical Institute 2017

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of WürzburgWürzburgGermany

Personalised recommendations