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On the estimation of nested Archimedean copulas: a theoretical and an experimental comparison

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  • Nathan Uyttendaele

    (Université catholique de Louvain)

Abstract

A lot of progress regarding estimation of nested Archimedean copulas has been booked since their introduction by Joe (Multivariate models and dependence concepts. Chapman and Hall, London, 1997). The currently published procedures can be seen as particular cases of two different, more general, approaches. In the first approach, the tree structure of the target nested Archimedean copulas is estimated using hierarchical clustering to get a binary tree, and then parts of this binary tree are collapsed according to some strategy. This two-step estimation of the tree structure paves the way for estimation of the generators according to the sufficient nesting condition afterwards, this sufficient nesting condition on the generators ensuring the resulting estimated nested Archimedean copula is a proper copula. In contrast to the first approach, the second approach estimates the tree structure free of any concern for the generators. While this is the main strength of this second approach, it is also its main weakness: estimation of the generators afterwards so that the resulting nested Archimedean copula is a proper copula still lacks a solution. In this paper, both approaches are formally explored, detailed explanations and examples are given, as well as results from a performance study where a new way of comparing tree structure estimators is offered. A nested Archimedean copula is also estimated based on exams results from 482 students, and a naive attempt to check the fit is made using principal component analysis.

Suggested Citation

  • Nathan Uyttendaele, 2018. "On the estimation of nested Archimedean copulas: a theoretical and an experimental comparison," Computational Statistics, Springer, vol. 33(2), pages 1047-1070, June.
    • Handle: RePEc:spr:compst:v:33:y:2018:i:2:d:10.1007_s00180-017-0743-1
    DOI: 10.1007/s00180-017-0743-1
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    References listed on IDEAS

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    1. Hofert, Marius & Pham, David, 2013. "Densities of nested Archimedean copulas," Journal of Multivariate Analysis, Elsevier, vol. 118(C), pages 37-52.
    2. Okhrin, Ostap & Ristig, Alexander, 2014. "Hierarchical Archimedean Copulae: The HAC Package," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 58(i04).
    3. Okhrin, Ostap & Okhrin, Yarema & Schmid, Wolfgang, 2013. "On the structure and estimation of hierarchical Archimedean copulas," Journal of Econometrics, Elsevier, vol. 173(2), pages 189-204.
    4. Ostap Okhrin & Alexander Ristig & Jeffrey Sheen & Stefan Trück, 2015. "Conditional Systemic Risk with Penalized Copula," SFB 649 Discussion Papers SFB649DP2015-038, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    5. Hofert, Marius & Maechler, Martin, 2011. "Nested Archimedean Copulas Meet R: The nacopula Package," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 39(i09).
    6. Segers, Johan & Uyttendaele, Nathan, 2014. "Nonparametric estimation of the tree structure of a nested Archimedean copula," Computational Statistics & Data Analysis, Elsevier, vol. 72(C), pages 190-204.
    7. Rezapour, Mohsen, 2015. "On the construction of nested Archimedean copulas for d-monotone generators," Statistics & Probability Letters, Elsevier, vol. 101(C), pages 21-32.
    8. Hofert, Marius, 2011. "Efficiently sampling nested Archimedean copulas," Computational Statistics & Data Analysis, Elsevier, vol. 55(1), pages 57-70, January.
    9. Andreas Alfons & Christophe Croux & Peter Filzmoser, 2017. "Robust Maximum Association Estimators," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(517), pages 436-445, January.
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    Cited by:

    1. Mai Jan-Frederik, 2019. "Simulation algorithms for hierarchical Archimedean copulas beyond the completely monotone case," Dependence Modeling, De Gruyter, vol. 7(1), pages 202-214, January.

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