IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v28y2015i4d10.1007_s10959-014-0541-4.html
   My bibliography  Save this article

Conditioning-based metrics on the space of multivariate copulas and their interrelation with uniform and levelwise convergence and Iterated Function Systems

Author

Listed:
  • Juan Fernández Sánchez

    (Universidad de Almería)

  • Wolfgang Trutschnig

    (University Salzburg)

Abstract

Using the one-to-one correspondence between copulas and special Markov kernels three strong metrics on the class $$\mathcal {C}_\rho $$ C ρ of $$\rho $$ ρ -dimensional copulas with $$\rho \ge 3$$ ρ ≥ 3 are studied. Being natural extensions of the two-dimensional versions introduced by Trutschnig (J Math Anal Appl 384:690–705, 2011), these metrics exhibit various good properties. In particular, it can be shown that the resulting metric spaces are separable and complete, which, as by-product, offers a simple separable and complete metrization of the so-called $$\partial $$ ∂ -convergence studied by Mikusinski and Taylor (Ann Polon Math 96:75–95, 2009, Metrika 72:385–414, 2010). As an additional consequence of completeness, it is proved that the construction of singular copulas with fractal support via special Iterated Function Systems also converges with respect to any of the three introduced metrics. Moreover, the interrelation with the uniform metric $$d_\infty $$ d ∞ is studied and convergence with respect to $$d_\infty $$ d ∞ is characterized in terms of level-set and endograph convergence with respect to the Hausdorff metric.

Suggested Citation

  • Juan Fernández Sánchez & Wolfgang Trutschnig, 2015. "Conditioning-based metrics on the space of multivariate copulas and their interrelation with uniform and levelwise convergence and Iterated Function Systems," Journal of Theoretical Probability, Springer, vol. 28(4), pages 1311-1336, December.
  • Handle: RePEc:spr:jotpro:v:28:y:2015:i:4:d:10.1007_s10959-014-0541-4
    DOI: 10.1007/s10959-014-0541-4
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-014-0541-4
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s10959-014-0541-4?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Vaart,A. W. van der, 2000. "Asymptotic Statistics," Cambridge Books, Cambridge University Press, number 9780521784504, October.
    2. Fredricks, Gregory A. & Nelsen, Roger B. & Rodriguez-Lallena, Jose Antonio, 2005. "Copulas with fractal supports," Insurance: Mathematics and Economics, Elsevier, vol. 37(1), pages 42-48, August.
    3. Elena Di Bernardino & Thomas Laloë & Véronique Maume-Deschamps & Clémentine Prieur, 2013. "Plug-in estimation of level sets in a non-compact setting with applications in multivariate risk theory," Post-Print hal-00580624, HAL.
    4. Durante, Fabrizio & Sánchez, Juan Fernández, 2012. "On the approximation of copulas via shuffles of Min," Statistics & Probability Letters, Elsevier, vol. 82(10), pages 1761-1767.
    5. So-Hsiang Chou & Truc T. Nguyen, 1990. "On Fréchet theorem in the set of measure preserving functions over the unit interval," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 13, pages 1-6, January.
    6. Durante, Fabrizio & Fernández-Sánchez, Juan, 2010. "Multivariate shuffles and approximation of copulas," Statistics & Probability Letters, Elsevier, vol. 80(23-24), pages 1827-1834, December.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Henryk Zähle, 2022. "A concept of copula robustness and its applications in quantitative risk management," Finance and Stochastics, Springer, vol. 26(4), pages 825-875, October.
    2. Griessenberger Florian & Trutschnig Wolfgang, 2022. "Maximal asymmetry of bivariate copulas and consequences to measures of dependence," Dependence Modeling, De Gruyter, vol. 10(1), pages 245-269, January.
    3. Sánchez Juan Fernández & Trutschnig Wolfgang, 2023. "A link between Kendall’s τ, the length measure and the surface of bivariate copulas, and a consequence to copulas with self-similar support," Dependence Modeling, De Gruyter, vol. 11(1), pages 1-14, January.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Durante, Fabrizio & Fernández Sánchez, Juan & Sempi, Carlo, 2013. "Multivariate patchwork copulas: A unified approach with applications to partial comonotonicity," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 897-905.
    2. Pfeifer Dietmar & Mändle Andreas & Ragulina Olena, 2017. "New copulas based on general partitions-of-unity and their applications to risk management (part II)," Dependence Modeling, De Gruyter, vol. 5(1), pages 246-255, October.
    3. Kasy, Maximilian, 2011. "A nonparametric test for path dependence in discrete panel data," Economics Letters, Elsevier, vol. 113(2), pages 172-175.
    4. Atı̇la Abdulkadı̇roğlu & Joshua D. Angrist & Yusuke Narita & Parag Pathak, 2022. "Breaking Ties: Regression Discontinuity Design Meets Market Design," Econometrica, Econometric Society, vol. 90(1), pages 117-151, January.
    5. Ashesh Rambachan & Jonathan Roth, 2020. "Design-Based Uncertainty for Quasi-Experiments," Papers 2008.00602, arXiv.org, revised Oct 2024.
    6. Debashis Ghosh, 2004. "Semiparametric methods for the binormal model with multiple biomarkers," The University of Michigan Department of Biostatistics Working Paper Series 1046, Berkeley Electronic Press.
    7. Brian D. Williamson & Peter B. Gilbert & Marco Carone & Noah Simon, 2021. "Nonparametric variable importance assessment using machine learning techniques," Biometrics, The International Biometric Society, vol. 77(1), pages 9-22, March.
    8. Fuchs Sebastian & McCord Yann, 2019. "On the lower bound of Spearman’s footrule," Dependence Modeling, De Gruyter, vol. 7(1), pages 126-132, January.
    9. Arie Beresteanu & Francesca Molinari, 2008. "Asymptotic Properties for a Class of Partially Identified Models," Econometrica, Econometric Society, vol. 76(4), pages 763-814, July.
    10. Laurent Davezies & Xavier D'Haultfoeuille & Yannick Guyonvarch, 2018. "Asymptotic results under multiway clustering," Papers 1807.07925, arXiv.org, revised Aug 2018.
    11. Dominic Edelmann & Tobias Terzer & Donald Richards, 2021. "A Basic Treatment of the Distance Covariance," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(1), pages 12-25, May.
    12. A Stefano Caria & Grant Gordon & Maximilian Kasy & Simon Quinn & Soha Osman Shami & Alexander Teytelboym, 2024. "An Adaptive Targeted Field Experiment: Job Search Assistance for Refugees in Jordan," Journal of the European Economic Association, European Economic Association, vol. 22(2), pages 781-836.
    13. Clément de Chaisemartin & Xavier D'Haultfœuille, 2020. "Two-Way Fixed Effects Estimators with Heterogeneous Treatment Effects," American Economic Review, American Economic Association, vol. 110(9), pages 2964-2996, September.
    14. Abe, Toshihiro & Miyata, Yoichi & Shiohama, Takayuki, 2023. "Bayesian estimation for mode and anti-mode preserving circular distributions," Econometrics and Statistics, Elsevier, vol. 27(C), pages 136-160.
    15. Bryan S. Graham, 2017. "An econometric model of network formation with degree heterogeneity," CeMMAP working papers 08/17, Institute for Fiscal Studies.
    16. Benoumechiara Nazih & Bousquet Nicolas & Michel Bertrand & Saint-Pierre Philippe, 2020. "Detecting and modeling critical dependence structures between random inputs of computer models," Dependence Modeling, De Gruyter, vol. 8(1), pages 263-297, January.
    17. Zhang, Feipeng & Xu, Yixiong & Fan, Caiyun, 2023. "Nonparametric inference of expectile-based value-at-risk for financial time series with application to risk assessment," International Review of Financial Analysis, Elsevier, vol. 90(C).
    18. Arun G. Chandrasekhar & Victor Chernozhukov & Francesca Molinari & Paul Schrimpf, 2019. "Best Linear Approximations to Set Identified Functions: With an Application to the Gender Wage Gap," NBER Working Papers 25593, National Bureau of Economic Research, Inc.
    19. Yuan Liao & Anna Simoni, 2012. "Semi-parametric Bayesian Partially Identified Models based on Support Function," Papers 1212.3267, arXiv.org, revised Nov 2013.
    20. Kline, Patrick, 2014. "A note on variance estimation for the Oaxaca estimator of average treatment effects," Economics Letters, Elsevier, vol. 122(3), pages 428-431.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jotpro:v:28:y:2015:i:4:d:10.1007_s10959-014-0541-4. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.