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Densities of nested Archimedean copulas

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  • Hofert, Marius
  • Pham, David

Abstract

Nested Archimedean copulas recently gained interest since they generalize the well-known class of Archimedean copulas to allow for partial asymmetry. Sampling algorithms and strategies have been well investigated for nested Archimedean copulas. However, for likelihood based inference it is important to have the density. The present work fills this gap. A general formula for the derivatives of the nodes and inner generators appearing in nested Archimedean copulas is developed. This leads to a tractable formula for the density of nested Archimedean copulas in arbitrary dimensions if the number of nesting levels is not too large. Various examples including famous Archimedean families and transformations of such are given. Furthermore, a numerically efficient way to evaluate the log-density is presented.

Suggested Citation

  • Hofert, Marius & Pham, David, 2013. "Densities of nested Archimedean copulas," Journal of Multivariate Analysis, Elsevier, vol. 118(C), pages 37-52.
    • Handle: RePEc:eee:jmvana:v:118:y:2013:i:c:p:37-52
    DOI: 10.1016/j.jmva.2013.03.006
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    References listed on IDEAS

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    1. Genest, Christian & Rémillard, Bruno & Beaudoin, David, 2009. "Goodness-of-fit tests for copulas: A review and a power study," Insurance: Mathematics and Economics, Elsevier, vol. 44(2), pages 199-213, April.
    2. 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).
    3. Hofert, Marius, 2011. "Efficiently sampling nested Archimedean copulas," Computational Statistics & Data Analysis, Elsevier, vol. 55(1), pages 57-70, January.
    4. Hofert, Marius & Mächler, Martin & McNeil, Alexander J., 2012. "Likelihood inference for Archimedean copulas in high dimensions under known margins," Journal of Multivariate Analysis, Elsevier, vol. 110(C), pages 133-150.
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    Cited by:

    1. Rezapour, Mohsen, 2015. "On the construction of nested Archimedean copulas for d-monotone generators," Statistics & Probability Letters, Elsevier, vol. 101(C), pages 21-32.
    2. Benjamin Poignard & Jean-David Fermanian, 2022. "The finite sample properties of sparse M-estimators with pseudo-observations," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 74(1), pages 1-31, February.
    3. Fatima Palacios‐Rodriguez & Elena Di Bernardino & Melina Mailhot, 2023. "Smooth copula‐based generalized extreme value model and spatial interpolation for extreme rainfall in Central Eastern Canada," Environmetrics, John Wiley & Sons, Ltd., vol. 34(3), May.
    4. Mirza Nazmul Hasan & Roel Braekers, 2021. "Estimation of the association parameters in hierarchically clustered survival data by nested Archimedean copula functions," Computational Statistics, Springer, vol. 36(4), pages 2755-2787, December.
    5. Di Bernardino Elena & Rullière Didier, 2013. "On certain transformations of Archimedean copulas: Application to the non-parametric estimation of their generators," Dependence Modeling, De Gruyter, vol. 1(2013), pages 1-36, October.
    6. Benjamin Poignard & Jean-David Fermanian, 2019. "The finite sample properties of Sparse M-estimators with Pseudo-Observations," Working Papers 2019-01, Center for Research in Economics and Statistics.
    7. 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.
    8. Shahid Latif & Slobodan P. Simonovic, 2023. "Trivariate Probabilistic Assessments of the Compound Flooding Events Using the 3-D Fully Nested Archimedean (FNA) Copula in the Semiparametric Distribution Setting," Water Resources Management: An International Journal, Published for the European Water Resources Association (EWRA), Springer;European Water Resources Association (EWRA), vol. 37(4), pages 1641-1693, March.
    9. 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.
    10. Nabil Kazi-Tani & Didier Rullière, 2019. "On a construction of multivariate distributions given some multidimensional marginals," Post-Print hal-01575169, HAL.
    11. Grothe, Oliver & Hofert, Marius, 2015. "Construction and sampling of Archimedean and nested Archimedean Lévy copulas," Journal of Multivariate Analysis, Elsevier, vol. 138(C), pages 182-198.
    12. Vettori, Sabrina & Huser, Raphael & Segers, Johan & Genton, Marc, 2017. "Bayesian Clustering and Dimension Reduction in Multivariate Extremes," LIDAM Discussion Papers ISBA 2017017, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    13. De Backer, Mickael & El Ghouch, Anouar & Van Keilegom, Ingrid, 2016. "Semiparametric Copula Quantile Regression for Complete or Censored Data," LIDAM Discussion Papers ISBA 2016009, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    14. Elena Di Bernardino & Didier Rullière, 2015. "Estimation of multivariate critical layers: Applications to rainfall data," Post-Print hal-00940089, HAL.
    15. Mazo, Gildas & Uyttendaele, Nathan, 2016. "Building conditionally dependent parametric one-factor copulas," LIDAM Discussion Papers ISBA 2016004, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).

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