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Dependence and Order in Families of Archimedean Copulas

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  • Nelsen, Roger B.

Abstract

The copula for a bivariate distribution functionH(x, y) with marginal distribution functionsF(x) andG(y) is the functionCdefined byH(x, y)=C(F(x), G(y)).Cis called Archimedean ifC(u, v)=[phi]-1([phi](u)+[phi](v)), where[phi]is a convex decreasing continuous function on (0, 1] with[phi](1)=0. A copula has lower tail dependence ifC(u, u)/uconverges to a constant[gamma]in (0, 1] asu-->0+; and has upper tail dependence ifC(u, u)/(1-u) converges to a constant[delta]in (0, 1] asu-->1-whereCdenotes the survival function corresponding toC. In this paper we develop methods for generating families of Archimedean copulas with arbitrary values of[gamma]and[delta], and present extensions to higher dimensions. We also investigate limiting cases and the concordance ordering of these families. In the process, we present answers to two open problems posed by Joe (1993,J. Multivariate Anal.46262-282).

Suggested Citation

  • Nelsen, Roger B., 1997. "Dependence and Order in Families of Archimedean Copulas," Journal of Multivariate Analysis, Elsevier, vol. 60(1), pages 111-122, January.
    • Handle: RePEc:eee:jmvana:v:60:y:1997:i:1:p:111-122
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    Cited by:

    1. Mesfioui, Mhamed & Quessy, Jean-François, 2008. "Dependence structure of conditional Archimedean copulas," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 372-385, March.
    2. Fadal Abdullah-A Aldhufairi & Ranadeera G.M. Samanthi & Jungsywan H. Sepanski, 2020. "New Families of Bivariate Copulas via Unit Lomax Distortion," Risks, MDPI, vol. 8(4), pages 1-19, October.
    3. Cooray Kahadawala, 2018. "Strictly Archimedean copulas with complete association for multivariate dependence based on the Clayton family," Dependence Modeling, De Gruyter, vol. 6(1), pages 1-18, February.
    4. Włodzimierz Wysocki, 2015. "Kendall's tau and Spearman's rho for n -dimensional Archimedean copulas and their asymptotic properties," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 27(4), pages 442-459, December.
    5. A. Sancetta & Satchell, S.E., 2001. "Bernstein Approximations to the Copula Function and Portfolio Optimization," Cambridge Working Papers in Economics 0105, Faculty of Economics, University of Cambridge.
    6. Quessy, Jean-François & Bahraoui, Tarik, 2014. "Weak convergence of empirical and bootstrapped C-power processes and application to copula goodness-of-fit," Journal of Multivariate Analysis, Elsevier, vol. 129(C), pages 16-36.
    7. Matthew Ames & Guillaume Bagnarosa & Gareth W. Peters, 2013. "Reinvestigating the Uncovered Interest Rate Parity Puzzle via Analysis of Multivariate Tail Dependence in Currency Carry Trades," Papers 1303.4314, arXiv.org, revised Jan 2014.
    8. Frahm, Gabriel, 2006. "On the extremal dependence coefficient of multivariate distributions," Statistics & Probability Letters, Elsevier, vol. 76(14), pages 1470-1481, August.
    9. Huang, Yu & Zhang, Bingzhe & Pang, Huizhen & Wang, Biao & Lee, Kwang Y. & Xie, Jiale & Jin, Yupeng, 2022. "Spatio-temporal wind speed prediction based on Clayton Copula function with deep learning fusion," Renewable Energy, Elsevier, vol. 192(C), pages 526-536.
    10. Mulero, Julio & Pellerey, Franco & Rodríguez-Griñolo, Rosario, 2010. "Stochastic comparisons for time transformed exponential models," Insurance: Mathematics and Economics, Elsevier, vol. 46(2), pages 328-333, April.
    11. Huang, Jen-Jsung & Lee, Kuo-Jung & Liang, Hueimei & Lin, Wei-Fu, 2009. "Estimating value at risk of portfolio by conditional copula-GARCH method," Insurance: Mathematics and Economics, Elsevier, vol. 45(3), pages 315-324, December.
    12. 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.
    13. Aleš Kresta, 2015. "Application of Performance Ratios in Portfolio Optimization," Acta Universitatis Agriculturae et Silviculturae Mendelianae Brunensis, Mendel University Press, vol. 63(6), pages 1969-1977.
    14. Okhrin Ostap & Okhrin Yarema & Schmid Wolfgang, 2013. "Properties of hierarchical Archimedean copulas," Statistics & Risk Modeling, De Gruyter, vol. 30(1), pages 21-54, March.
    15. Hanif, Waqas & Mensi, Walid & Alomari, Mohammad & Andraz, Jorge Miguel, 2023. "Downside and upside risk spillovers between precious metals and currency markets: Evidence from before and during the COVID-19 crisis," Resources Policy, Elsevier, vol. 81(C).
    16. Qiang Liu & Aiping Tang & Zhongyue Wang & Buyue Zhao, 2023. "Exploring the road icing risk: considering the dependence of icing-inducing factors," Natural Hazards: Journal of the International Society for the Prevention and Mitigation of Natural Hazards, Springer;International Society for the Prevention and Mitigation of Natural Hazards, vol. 115(3), pages 2161-2178, February.

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