IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v117y2013icp246-256.html
   My bibliography  Save this article

Homogeneous distributions—And a spectral representation of classical mean values and stable tail dependence functions

Author

Listed:
  • Ressel, Paul

Abstract

Homogeneous distributions on R+d and on R¯+d∖︀{∞¯d} are shown to be Bauer simplices when normalized. This is used to provide spectral representations for the classical power mean values Mt(x) which turn out to be unique mixtures of the functions x⟼mini≤d(aixi) for t≤1 (with some gaps depending on the dimension d), resp. x⟼maxi≤d(aixi) for t≥1 (without gaps), where ai≥0.

Suggested Citation

  • Ressel, Paul, 2013. "Homogeneous distributions—And a spectral representation of classical mean values and stable tail dependence functions," Journal of Multivariate Analysis, Elsevier, vol. 117(C), pages 246-256.
    • Handle: RePEc:eee:jmvana:v:117:y:2013:i:c:p:246-256
    DOI: 10.1016/j.jmva.2013.02.013
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047259X13000274
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.jmva.2013.02.013?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. Ressel, Paul, 2011. "Monotonicity properties of multivariate distribution and survival functions -- With an application to Lévy-frailty copulas," Journal of Multivariate Analysis, Elsevier, vol. 102(3), pages 393-404, March.
    2. Genest, Christian & Rivest, Louis-Paul, 1989. "A characterization of gumbel's family of extreme value distributions," Statistics & Probability Letters, Elsevier, vol. 8(3), pages 207-211, August.
    3. Ressel, Paul, 2012. "Functions operating on multivariate distribution and survival functions—With applications to classical mean-values and to copulas," Journal of Multivariate Analysis, Elsevier, vol. 105(1), pages 55-67.
    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. Elena Di Bernardino & Didier Rullière, 2016. "On an asymmetric extension of multivariate Archimedean copulas based on quadratic form," Working Papers hal-01147778, HAL.
    2. Mercadier Cécile & Ressel Paul, 2021. "Hoeffding–Sobol decomposition of homogeneous co-survival functions: from Choquet representation to extreme value theory application," Dependence Modeling, De Gruyter, vol. 9(1), pages 179-198, January.
    3. Einmahl, John & Kiriliouk, A. & Segers, J.J.J., 2016. "A Continuous Updating Weighted Least Squares Estimator of Tail Dependence in High Dimensions," Other publications TiSEM a3e7350b-4773-4bd8-9c3c-6, Tilburg University, School of Economics and Management.
    4. Molchanov, Ilya & Strokorb, Kirstin, 2016. "Max-stable random sup-measures with comonotonic tail dependence," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2835-2859.
    5. Ressel Paul, 2018. "A multivariate version of Williamson’s theorem, ℓ1-symmetric survival functions, and generalized Archimedean copulas," Dependence Modeling, De Gruyter, vol. 6(1), pages 356-368, December.
    6. Mai, Jan-Frederik & Scherer, Matthias, 2020. "On the structure of exchangeable extreme-value copulas," Journal of Multivariate Analysis, Elsevier, vol. 180(C).
    7. Charpentier, A. & Fougères, A.-L. & Genest, C. & Nešlehová, J.G., 2014. "Multivariate Archimax copulas," Journal of Multivariate Analysis, Elsevier, vol. 126(C), pages 118-136.
    8. Kiriliouk, Anna & Segers, Johan & Tafakori, Laleh, 2018. "An estimator of the stable tail dependence function based on the empirical beta copula," LIDAM Discussion Papers ISBA 2018029, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    9. Hofert, Marius & Huser, Raphaël & Prasad, Avinash, 2018. "Hierarchical Archimax copulas," Journal of Multivariate Analysis, Elsevier, vol. 167(C), pages 195-211.
    10. Górecki, Jan & Hofert, Marius & Okhrin, Ostap, 2021. "Outer power transformations of hierarchical Archimedean copulas: Construction, sampling and estimation," Computational Statistics & Data Analysis, Elsevier, vol. 155(C).
    11. Bernhart German & Scherer Matthias & Mai Jan-Frederik, 2015. "On the construction of low-parametric families of min-stable multivariate exponential distributions in large dimensions," Dependence Modeling, De Gruyter, vol. 3(1), pages 1-18, May.
    12. Ressel Paul, 2022. "Stable tail dependence functions – some basic properties," Dependence Modeling, De Gruyter, vol. 10(1), pages 225-235, January.
    13. Ressel Paul, 2019. "Copulas, stable tail dependence functions, and multivariate monotonicity," Dependence Modeling, De Gruyter, vol. 7(1), pages 247-258, January.
    14. Rootzen, Holger & Segers, Johan & Wadsworth, Jennifer, 2017. "Multivariate generalized Pareto distributions: parametrizations, representations, and properties," LIDAM Discussion Papers ISBA 2017016, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    15. Mai Jan-Frederik, 2020. "The de Finetti structure behind some norm-symmetric multivariate densities with exponential decay," Dependence Modeling, De Gruyter, vol. 8(1), pages 210-220, January.
    16. Einmahl, John & Segers, Johan, 2020. "Empirical Tail Copulas for Functional Data," Other publications TiSEM edc722e6-cc70-4221-87a2-8, Tilburg University, School of Economics and Management.
    17. Hofert, Marius, 2021. "Right-truncated Archimedean and related copulas," Insurance: Mathematics and Economics, Elsevier, vol. 99(C), pages 79-91.
    18. Kiriliouk, Anna & Segers, Johan & Tafakori, Laleh, 2017. "An estimator of the stable tail dependence function based on the empirical beta copula," LIDAM Discussion Papers ISBA 2017028, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    19. Mhalla, Linda & Chavez-Demoulin, Valérie & Naveau, Philippe, 2017. "Non-linear models for extremal dependence," Journal of Multivariate Analysis, Elsevier, vol. 159(C), pages 49-66.
    20. Mai Jan-Frederik, 2020. "The de Finetti structure behind some norm-symmetric multivariate densities with exponential decay," Dependence Modeling, De Gruyter, vol. 8(1), pages 210-220, January.
    21. Durante Fabrizio & Sánchez Juan Fernández & Sempi Carlo, 2018. "A note on bivariate Archimax copulas," Dependence Modeling, De Gruyter, vol. 6(1), pages 178-182, October.
    22. Marcon, Giulia & Padoan, Simone & Naveau, Philippe & Muliere, Pietro & Segers, Johan, 2016. "Multivariate Nonparametric Estimation of the Pickands Dependence Function using Bernstein Polynomials," LIDAM Discussion Papers ISBA 2016020, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    23. Rootzén, Holger & Segers, Johan & Wadsworth, Jennifer L., 2018. "Multivariate generalized Pareto distributions: Parametrizations, representations, and properties," Journal of Multivariate Analysis, Elsevier, vol. 165(C), pages 117-131.
    24. Kiriliouk, Anna, 2017. "Hypothesis testing for tail dependence parameters on the boundary of the parameter space with application to generalized max-linear models," LIDAM Discussion Papers ISBA 2017027, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    25. Mai, Jan-Frederik, 2018. "Extreme-value copulas associated with the expected scaled maximum of independent random variables," Journal of Multivariate Analysis, Elsevier, vol. 166(C), pages 50-61.

    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. Ressel Paul, 2022. "Stable tail dependence functions – some basic properties," Dependence Modeling, De Gruyter, vol. 10(1), pages 225-235, January.
    2. Ressel Paul, 2019. "Copulas, stable tail dependence functions, and multivariate monotonicity," Dependence Modeling, De Gruyter, vol. 7(1), pages 247-258, January.
    3. Ressel Paul, 2018. "A multivariate version of Williamson’s theorem, ℓ1-symmetric survival functions, and generalized Archimedean copulas," Dependence Modeling, De Gruyter, vol. 6(1), pages 356-368, December.
    4. Ressel Paul, 2023. "Functions operating on several multivariate distribution functions," Dependence Modeling, De Gruyter, vol. 11(1), pages 1-11, January.
    5. Okhrin Ostap & Okhrin Yarema & Schmid Wolfgang, 2013. "Properties of hierarchical Archimedean copulas," Statistics & Risk Modeling, De Gruyter, vol. 30(1), pages 21-54, March.
    6. Mercadier Cécile & Ressel Paul, 2021. "Hoeffding–Sobol decomposition of homogeneous co-survival functions: from Choquet representation to extreme value theory application," Dependence Modeling, De Gruyter, vol. 9(1), pages 179-198, January.
    7. Hashorva, Enkelejd & Rullière, Didier, 2020. "Asymptotic domination of sample maxima," Statistics & Probability Letters, Elsevier, vol. 160(C).
    8. Jan-Frederik Mai & Steffen Schenk & Matthias Scherer, 2017. "Two Novel Characterizations of Self-Decomposability on the Half-Line," Journal of Theoretical Probability, Springer, vol. 30(1), pages 365-383, March.
    9. Li, Haijun & Wu, Peiling, 2013. "Extremal dependence of copulas: A tail density approach," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 99-111.
    10. Mainik Georg & Rüschendorf Ludger, 2012. "Ordering of multivariate risk models with respect to extreme portfolio losses," Statistics & Risk Modeling, De Gruyter, vol. 29(1), pages 73-106, March.
    11. Bücher, Axel & Dette, Holger & Volgushev, Stanislav, 2012. "A test for Archimedeanity in bivariate copula models," Journal of Multivariate Analysis, Elsevier, vol. 110(C), pages 121-132.
    12. Mazo, Gildas & Girard, Stéphane & Forbes, Florence, 2015. "A class of multivariate copulas based on products of bivariate copulas," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 363-376.
    13. Matthieu Garcin & Maxime L. D. Nicolas, 2021. "Nonparametric estimator of the tail dependence coefficient: balancing bias and variance," Papers 2111.11128, arXiv.org, revised Jul 2023.
    14. Yong Ma & Zhengjun Zhang & Weiguo Zhang & Weidong Xu, 2015. "Evaluating the Default Risk of Bond Portfolios with Extreme Value Theory," Computational Economics, Springer;Society for Computational Economics, vol. 45(4), pages 647-668, April.
    15. Charpentier, A. & Fougères, A.-L. & Genest, C. & Nešlehová, J.G., 2014. "Multivariate Archimax copulas," Journal of Multivariate Analysis, Elsevier, vol. 126(C), pages 118-136.
    16. Tankov, Peter, 2016. "Tails of weakly dependent random vectors," Journal of Multivariate Analysis, Elsevier, vol. 145(C), pages 73-86.
    17. Shenkman, Natalia, 2017. "A natural parametrization of multivariate distributions with limited memory," Journal of Multivariate Analysis, Elsevier, vol. 155(C), pages 234-251.
    18. Segers, J.J.J., 2004. "Non-Parametric Inference for Bivariate Extreme-Value Copulas," Discussion Paper 2004-91, Tilburg University, Center for Economic Research.
    19. Molchanov, Ilya & Strokorb, Kirstin, 2016. "Max-stable random sup-measures with comonotonic tail dependence," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2835-2859.
    20. Segers, J.J.J., 2004. "Non-Parametric Inference for Bivariate Extreme-Value Copulas," Other publications TiSEM 3e837d24-e733-407c-bfaa-f, Tilburg University, School of Economics and Management.

    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:eee:jmvana:v:117:y:2013:i:c:p:246-256. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    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.