Skip to main content
SpringerLink
Log in

A Small-Sample Nonparametric Independence Test for the Archimedean Family of Bivariate Copulas

  • Conference paper
Soft Methods for Handling Variability and Imprecision

Part of the book series: Advances in Soft Computing ((AINSC,volume 48))

Abstract

In this paper we study the problem of independence of two continuous random variables using the fact that there exists a unique copula that characterizes independence, and that such copula is of Archimedean type. We use properties of the empirical diagonal to build nonparametric independence tests for small samples, under the assumption that the underlying copula belongs to the Archimedean family, giving solution to an open problem proposed by Alsina et al.[2].

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 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
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.

References

  1. Aguiló, I., Suñer, J., Torrens, J.: Matrix representation of discrete quasi-copulas. Fuzzy Sets Syst. (in press, 2008) doi:10.1016/j.fss2007.10.004

    Google Scholar 

  2. Alsina, C., Frank, M.J., Schweizer, B.: Problems on associative functions. Aequationes Math. 66, 128–140 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  3. Alsina, C., Frank, M.J., Schweizer, B.: Associative Functions: Triangular Norms and Copulas. World Scientific Publishing Co, Singapore (2006)

    MATH  Google Scholar 

  4. Anderson, T.W., Darling, D.A.: A test of goodness of fit. J. Amer. Statist. Assoc. 49, 765–769 (1954)

    Article  MATH  MathSciNet  Google Scholar 

  5. Blum, J.R., Kiefer, J., Rosenblatt, M.: Distribution-free tests of independence based on the sample distribution function. Ann. Math. Statist. 32, 485–498 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  6. Darsow, W.F., Frank, M.J.: Associative functions and Abel-Schröder systems. Publ. Math. Debrecen. 30, 253–272 (1983)

    MATH  MathSciNet  Google Scholar 

  7. Erdely, A.: Diagonal Properties of the Empirical Copula and Applications. Construction of Families of Copulas with Given Restrictions. Ph.D. Dissertation, Universidad Nacional Autónoma de México (2007)

    Google Scholar 

  8. Erdely, A., González-Barrios, J.M.: On the construction of families of absolutely continuous copulas with given restrictions. Comm. Statist. Theory Methods 35, 649–659 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Frank, M.J.: Diagonals of copulas and Schröder’s equation. Aequationes Math. 51, 150 (1996)

    Google Scholar 

  10. Fredricks, G.A., Nelsen, R.B.: Copulas constructed from diagonal sections. In: Beneš, V., Štěpán, J. (eds.) Distributions with Given Marginals and Moment Problems, pp. 129–136. Kluwer Academic Publishers, Dordrecht (1997)

    Google Scholar 

  11. Hollander, M., Wolfe, D.A.: Nonparametric Statistical Methods. Wiley-Interscience, New York (1999)

    MATH  Google Scholar 

  12. Kallenberg, W.C.M., Ledwina, T.: Data-driven Rank tests for independence. J. Amer. Stat. Assoc. 94, 285–301 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kolesárová, A., Mesiar, R., Mordelová, J., Sempi, C.: Discrete Copulas. IEEE Trans. Fuzzy Syst. 14, 698–705 (2006)

    Article  Google Scholar 

  14. Kolesárová, A., Mordelová, J.: Quasi-copulas and copulas on a discrete scale. Soft Computing 10, 495–501 (2006)

    Article  MATH  Google Scholar 

  15. Mayor, G., Suñer, J., Torrens, J.: Copula-like operations on finite settings. IEEE Trans. Fuzzy Syst. 13, 468–477 (2005)

    Article  Google Scholar 

  16. Mesiar, R.: Discrete copulas - what they are. In: Montseny, E., Sobrevilla, P. (eds.) Joint EUSFLAT-LFA 2005 Conference Proceedings. Universitat Politecnica de Catalunya, Barcelona, pp. 927-930 (2005)

    Google Scholar 

  17. Nelsen, R.B.: An Introduction to Copulas, 2nd edn. Springer, New York (2006)

    MATH  Google Scholar 

  18. Nelsen, R.B., Quesada-Molina, J.J., Rodríguez-Lallena, J.A., Úbeda-Flores, M.: On the construction of copulas and quasi-copulas with given diagonal sections. Insurance Math. Econom. 42, 473–483 (2008)

    Article  MathSciNet  Google Scholar 

  19. Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. North-Holland, New York (1983)

    MATH  Google Scholar 

  20. Sklar, A.: Fonctions de répartition à n dimensions et leurs marges. Publ. Inst.Statist. Univ. Paris 8, 229–231 (1959)

    MathSciNet  Google Scholar 

  21. Sungur, E.A., Yang, Y.: Diagonal copulas of Archimedean class. Comm. Statist. Theory Methods 25, 1659–1676 (1996)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Escuela de Actuaría, Universidad Anáhuac México Norte, Huixquilucan, México

    Arturo Erdely

  2. Departamento de Probabilidad y Estadística, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, México D.F., México

    José M. González-Barrios

Authors
  1. Arturo Erdely
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. José M. González-Barrios
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Université Paul Sabatier, Toulouse, France

    Didier Dubois  & Henri Prade  & 

  2. Universidad de Oviedo, Spain

    M. Asunción Lubiano  & María Ángeles Gil  & 

  3. Polish Academy of Sciences, Warsaw, Poland

    Przemysław Grzegorzewski

  4. Polish Academy of Sciences,, Warsaw, Poland

    Olgierd Hryniewicz

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Erdely, A., González-Barrios, J.M. (2008). A Small-Sample Nonparametric Independence Test for the Archimedean Family of Bivariate Copulas. In: Dubois, D., Lubiano, M.A., Prade, H., Gil, M.Á., Grzegorzewski, P., Hryniewicz, O. (eds) Soft Methods for Handling Variability and Imprecision. Advances in Soft Computing, vol 48. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85027-4_15

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-85027-4_15

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-85026-7

  • Online ISBN: 978-3-540-85027-4

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation