Nothing Special   »   [go: up one dir, main page]

Skip to main content

How Many Archimedean Copulæ Are There?

  • Conference paper
Advances in Computational Intelligence (IPMU 2012)

Abstract

Two algebraic notions, power of an associative binary function and nilpotency, are used in order to show that every bivariate Archimedean copula C is isomorphic to either the independence copula Π2, if it is strict, or to the lower Fréchet–Hoeffding bound W 2, if it is nilpotent.

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

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Alsina, C., Frank, M.J., Schweizer, B.: Associative functions. Triangular norms and copulas. World Scientific, Singapore (2006)

    Book  MATH  Google Scholar 

  2. Arnold, V.I.: Concerning nthe representability of functions of two variables in the form χ(ϕ(x) + ψ(y)). Uspehi Mat. Nauk 12, 119–121 (1957)

    MathSciNet  Google Scholar 

  3. Barnett, V.: Some bivariate uniform distributions. Comm. Statist. A—Theory Methods 9, 453–461 (1980)

    Article  MathSciNet  Google Scholar 

  4. Frank, M.J.: On the simultaneous associativity of F(x,y) and x + y − F(x,y). Aequationes Math. 19, 194–226 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  5. Genest, C., Ghoudi, K.: Une famille de lois bidimensionnelle insolite. C. R. Acad. Sci Paris Sér. I Math. 318, 351–354 (1994)

    MathSciNet  MATH  Google Scholar 

  6. Genest, C., MacKay, J.: The joy of copulas: bivariate distributions with uniform marginals. Amer. Statist. 40, 280–283 (1986)

    MathSciNet  Google Scholar 

  7. Genest, C., MacKay, J.: Copules archimédienes et familles de lois bidimensionnelles dont les marges sont données. Canad. J. Statist. 14, 280–283 (1986)

    Article  MathSciNet  Google Scholar 

  8. Grabisch, M., Marichal, J.–L., Mesiar, R., Pap, E.: Aggregation functions. Encyclopedia of Mathematics and its Applications, vol. 127. Cambridge Univesity Press, New York (2009)

    MATH  Google Scholar 

  9. Gumbel, E.J.: Distributions à plusieurs variables dont les marges sont données. C. R. Acad. Sci. Paris 246, 2717–2719 (1958)

    MathSciNet  MATH  Google Scholar 

  10. Gumbel, E.J.: Distributions à plusieurs variables dont les marges sont données. C. R. Acad. Sci. Paris 246, 2717–2719 (1960)

    MathSciNet  Google Scholar 

  11. Hamacher, H.: Über logische Aggregationen nicht–binär explizierter Entscheidungskriterien. Rita G. Fischer Verlag (1978)

    Google Scholar 

  12. Hougaard, P.: A family of multivariate failure time distributions. Biometrika 73, 671–678 (1986)

    MathSciNet  MATH  Google Scholar 

  13. Hutchinson, T.P., Lai, C.D.: Continuous bivariate distributions. Emphasising applications. Rumsby Scientific Publishing, Adelaide (1990)

    MATH  Google Scholar 

  14. Joe, H.: Parametric families of multivariate distributions with given marginals. J. Multivariate Anal. 46, 262–282 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  15. Joe, H.: Multivariate models and dependence concepts. Chapman & Hall, London (1997)

    MATH  Google Scholar 

  16. Klement, E.P., Mesiar, R., Pap, E.: Triangular norms. Kluwer, Dordrecht (2000)

    MATH  Google Scholar 

  17. Krause, G.: A strengthened form of Ling’s theorem on associative functions. Ph. D. Dissertation, Illinois Institute of Technology (1981)

    Google Scholar 

  18. Ling, C.H.: Representation of associative functions. Publ. Math. Debrecen 12, 189–212 (1965)

    MathSciNet  Google Scholar 

  19. McNeil, A.J., Nešlehová, J.: Multivariate Archimedean copulas, d–monotone functions and l 1–norm symmetric distributions. Ann. Statist. 37, 3059–3097 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Michiels, F., De Schepper, A.: Understanding copula transforms: a review of dependence properties, Research paper 2009–012, Department of accounting and finance, University of Antwerp (2009)

    Google Scholar 

  21. Moynihan, R.: On τ T –semigroups of probability distribution functions II. Aequationes Math. 17, 19–40 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  22. Nelsen, R.B.: An introduction to copulas, 2nd edn. Lecture Notes in Statistics, vol. 139. Springer, New York (1999/2006)

    MATH  Google Scholar 

  23. Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. North-Holland, New York (1983); reprinted, Dover, Mineola (2005)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Sempi, C. (2012). How Many Archimedean Copulæ Are There?. In: Greco, S., Bouchon-Meunier, B., Coletti, G., Fedrizzi, M., Matarazzo, B., Yager, R.R. (eds) Advances in Computational Intelligence. IPMU 2012. Communications in Computer and Information Science, vol 298. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31715-6_21

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-31715-6_21

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-31714-9

  • Online ISBN: 978-3-642-31715-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics