Skip to main content
SpringerLink
Log in

On Quantifying and Estimating Directed Dependence

  • Conference paper
  • First Online:
Book cover Building Bridges between Soft and Statistical Methodologies for Data Science (SMPS 2022)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 1433))

Included in the following conference series:

  • 337 Accesses

Abstract

Considering that a (random) variable X may provide more information about a (random) variable Y than vice versa it is natural that dependence measures, i.e., notions quantifying the extent of dependence, are not necessarily symmetric. Working with Markov kernels (regular conditional distributions) allows to construct the measure which assigns every copula a value in [0, 1], which is 0 exactly in the case of independence, and 1 exclusively for Y being a function of X. More importantly, given samples of X and Y and considering so-called checkerboard estimators it is possible to derive a strongly consistent estimator for the dependence measure which also exhibits a good performance for small and medium sample sizes. After sketching the background on the dependence measure and its checkerboard estimator, and illustrating its performance in terms of a small simulation study we discuss how the studied approach can be generalized to the multivariate question of quantifying the extent of dependence of a random variable Y on an ensemble of random variables.

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 189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 249.99
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

References

  • Durante, F., Sempi, C.: Principles of Copula Theory. CRC Press, New York (2016)

    MATH  Google Scholar 

  • Griessenberger, F., Junker, R.R., Trutschnig, W.: On a multivariate copula-based dependence measure and its estimation. Electron. J. Stat. 16(1), 2206–2251 (2022)

    Article  MathSciNet  Google Scholar 

  • Janssen, P., Swanepoel, J., Ververbeke, N.: Large sample behaviour of the Bernstein estimator. J. Stat. Plann. Infer. 142, 1189–1197 (2012)

    Article  Google Scholar 

  • Junker, R.R., Griessenberger, F., Trutschnig, W.: Estimating scale-invariant directed dependence of bivariate distributions. Comput. Stat. Data Anal. 153, 107058 (2021)

    Article  MathSciNet  Google Scholar 

  • Kallenberg, O.: Foundations of Modern Probability. Springer, Heidelberg (1997)

    MATH  Google Scholar 

  • Klenke, A.: Wahrscheinlichkeitstheorie. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-77571-3

  • Li, H., Scarsini, M., Shaked, M.: Linkages: a tool for the construction of multivariate distributions with given nonoverlapping multivariate marginals. J. Multivar. Anal. 56(1), 20–41 (1996)

    Article  MathSciNet  Google Scholar 

  • Li, X., Mikusinski, P., Taylor, M.D.: Strong approximation of copulas. J. Math. Anal. Appl. 255, 608–623 (1998)

    Article  MathSciNet  Google Scholar 

  • Nelsen, R.B.: An Introduction to Copulas. Springer, New York (2006). https://doi.org/10.1007/0-387-28678-0

    Book  MATH  Google Scholar 

  • Schweizer, B., Wolff, E.F.: On nonparametric measures of dependence for random variables. Ann. Stat. 9(4), 879–885 (1981)

    Article  MathSciNet  Google Scholar 

  • Siburg, K.F., Stoimenov, P.F.: A measure of mutual complete dependence. Metrika 71(2), 239–251 (2010)

    Article  MathSciNet  Google Scholar 

  • Trutschnig, W.: On a strong metric on the space of copulas and its induced dependence measure. J. Math. Anal. Appl. 384, 690–705 (2011)

    Article  MathSciNet  Google Scholar 

  • Wiedermann, W., Kim, D., Sungur, E.A., von Eye, A.: Direction Dependence in Statistical Modeling: Methods of Analysis. Wiley, Hoboken (2020)

    Book  Google Scholar 

Download references

Acknowledgements

The first author gratefully acknowledges the support of the WISS 2025 project ‘IDA-lab Salzburg’ (20204-WISS/225/197-2019 & 20102-F1901166-KZP), the second author gratefully acknowledge the support of the Austrian FWF START project Y1102 ‘Successional Generation of Functional Multidiversity’.

Author information

Authors and Affiliations

  1. Department for Artificial Intelligence and Human Interfaces (AIHI), University of Salzburg, Hellbrunnerstraße 34, Salzburg, Austria

    Wolfgang Trutschnig & Florian Griessenberger

Authors
  1. Wolfgang Trutschnig
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Florian Griessenberger
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wolfgang Trutschnig .

Editor information

Editors and Affiliations

  1. Departamento de Estadística e I.O., University of Valladolid, Valladolid, Spain

    Luis A. García-Escudero

  2. Departamento de Estadística e I.O., University of Valladolid, Valladolid, Spain

    Alfonso Gordaliza

  3. Departamento de Estadística e I.O., University of Valladolid, Valladolid, Spain

    Agustín Mayo

  4. Departamento de Estadística, I.O. y D.M., University of Oviedo, Oviedo, Spain

    María Asunción Lubiano Gomez

  5. Departamento de Estadística, I.O. y D.M., University of Oviedo, Oviedo, Spain

    Maria Angeles Gil

  6. Department of Computational Statistics and Data Analysis, Warsaw University of Technology, Warsaw, Poland

    Przemyslaw Grzegorzewski

  7. Department of Stochastic Methods, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

    Olgierd Hryniewicz

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Trutschnig, W., Griessenberger, F. (2023). On Quantifying and Estimating Directed Dependence. In: García-Escudero, L.A., et al. Building Bridges between Soft and Statistical Methodologies for Data Science . SMPS 2022. Advances in Intelligent Systems and Computing, vol 1433. Springer, Cham. https://doi.org/10.1007/978-3-031-15509-3_50

Download citation

Publish with us

Policies and ethics

Search

Navigation