A Markov Kernel Approach to Multivariate Archimedean Copulas

Kasper, Thimo M.; Trutschnig, Wolfgang

doi:10.1007/978-3-031-15509-3_30

Thimo M. Kasper²¹ &
Wolfgang Trutschnig²¹

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

Included in the following conference series:

International Conference on Soft Methods in Probability and Statistics

362 Accesses
1 Citations

Abstract

This contribution deals with describing the mass distribution of multivariate Archimedean copulas via Markov kernels, i.e. regular conditional distributions. After establishing an explicit expression for the Markov kernel of an Archimedean copula we use it to provide alternative derivations of the formulas for the Kendall distribution function as well as for the mass of the level sets. The described approach is purely based on Markov kernels and does not build upon \(\ell _1\)-norm symmetric distributions which seem to be the current standard.

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)

eBook: USD 189.00; Price excludes VAT (USA)

Softcover Book: USD 249.99; Price excludes VAT (USA)

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

Charpentier, A., Segers, J.: Convergence of Archimedean copulas. Stat. Probab. Lett. 78, 412–419 (2008)
Article MathSciNet Google Scholar
Durante, F., Sempi, C.: Principles of Copula Theory. Taylor & Francis Group LLC, Boca Raton (2016)
MATH Google Scholar
Fernández-Sánchez, J., Trutschnig, W.: Singularity aspects of Archimedean copulas. J. Math. Anal. Appl. 432, 103–113 (2015)
Article MathSciNet Google Scholar
Genest, C., Rivest, L.: Statistical inference procedures for bivariate Archimedean copulas. J. Am. Stat. Assoc. 88, 1034–1043 (1993)
Article MathSciNet Google Scholar
Kallenberg, O.: Foundations of Modern Probability. Springer, New York (2002). https://doi.org/10.1007/978-3-030-61871-1
Book MATH Google Scholar
Kasper, T., Fuchs, S., Trutschnig, W.: On weak conditional convergence of bivariate Archimedean and extreme value copulas, and consequences to nonparametric estimation. Bernoulli 4(27), 2217–2240 (2021)
MathSciNet MATH Google Scholar
Rüschendorf, L.: Wahrscheinlichkeitstheorie. Masterclass Series. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-662-48937-6
McNeil, A., Nešlehová, J.: Multivariate Archimedean copulas d-monotone functions and \(\ell _1\)-norm symmetric distributions. Ann. Stat. 37, 3059–3097 (2009)
Article Google Scholar
Nelsen, R.: An Introduction to Copulas. Springer, Heidelberg (2006). https://doi.org/10.1007/0-387-28678-0
Book MATH Google Scholar

Download references

Acknowledgements

The first author gratefully acknowledges the financial support from Porsche Holding Austria and Land Salzburg within the WISS 2025 project ‘KFZ’ (P1900123). The second author gratefully acknowledges the support of the WISS 2025 project ‘IDA-lab Salzburg’ (20204-WISS/225/197-2019 and 20102-F1901166-KZP).

Author information

Authors and Affiliations

Department for Artificial Intelligence and Human Interfaces, University of Salzburg, Hellbrunnerstraße 34, 5020, Salzburg, Austria
Thimo M. Kasper & Wolfgang Trutschnig

Authors

Thimo M. Kasper
View author publications
You can also search for this author in PubMed Google Scholar
Wolfgang Trutschnig
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

Departamento de Estadística e I.O., University of Valladolid, Valladolid, Spain
Luis A. García-Escudero
Departamento de Estadística e I.O., University of Valladolid, Valladolid, Spain
Alfonso Gordaliza
Departamento de Estadística e I.O., University of Valladolid, Valladolid, Spain
Agustín Mayo
Departamento de Estadística, I.O. y D.M., University of Oviedo, Oviedo, Spain
María Asunción Lubiano Gomez
Departamento de Estadística, I.O. y D.M., University of Oviedo, Oviedo, Spain
Maria Angeles Gil
Department of Computational Statistics and Data Analysis, Warsaw University of Technology, Warsaw, Poland
Przemyslaw Grzegorzewski
Department of Stochastic Methods, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland
Olgierd Hryniewicz

Rights and permissions

Reprints and permissions

Copyright information

About this paper

Cite this paper

Kasper, T.M., Trutschnig, W. (2023). A Markov Kernel Approach to Multivariate Archimedean Copulas. 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_30

Download citation

DOI: https://doi.org/10.1007/978-3-031-15509-3_30
Published: 25 August 2022
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-15508-6
Online ISBN: 978-3-031-15509-3
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)

Publish with us

Policies and ethics

A Markov Kernel Approach to Multivariate Archimedean Copulas