IDEAS home Printed from https://ideas.repec.org/p/aiz/louvad/2023034.html
   My bibliography  Save this paper

Multivariate generalized Pareto distributions along extreme directions

Author

Listed:
  • Mourahib, Anas

    (Université catholique de Louvain, LIDAM/ISBA, Belgium)

  • Kiriliouk, Anna

    (UNamur)

  • Segers, Johan

    (Université catholique de Louvain, LIDAM/ISBA, Belgium)

Abstract

When modeling a vector of risk variables, extreme scenarios are often of special interest. The peaks-over-thresholds method hinges on the notion that, asymptotically, the excesses over a vector of high thresholds follow a multivariate generalized Pareto distribution. However, existing literature has primarily concentrated on the setting when all risk variables are always large simultaneously. In reality, this assumption is often not met, especially in high dimensions. In response to this limitation, we study scenarios where distinct groups of risk variables may exhibit joint extremes while others do not. These discernible groups are derived from the angular measure inherent in the corresponding max-stable distribution, whence the term extreme direction. We explore such extreme directions within the framework of multivariate generalized Pareto distributions, with a focus on their probability density functions in relation to an appropriate dominating measure. Furthermore, we provide a stochastic construction that allows any prespecified set of risk groups to constitute the distribution’s extreme directions. This construction takes the form of a smoothed max-linear model and accommodates the full spectrum of conceivable max-stable dependence structures. Additionally, we introduce a generic simulation algorithm tailored for multivariate generalized Pareto distributions, offering specific implementations for extensions of the logistic and Hüsler–Reiss families capable of carrying arbitrary extreme directions.

Suggested Citation

  • Mourahib, Anas & Kiriliouk, Anna & Segers, Johan, 2023. "Multivariate generalized Pareto distributions along extreme directions," LIDAM Discussion Papers ISBA 2023034, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
  • Handle: RePEc:aiz:louvad:2023034
    as

    Download full text from publisher

    File URL: https://dial.uclouvain.be/pr/boreal/en/object/boreal%3A280209/datastream/PDF_01/view
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Maël Chiapino & Stephan Clémençon & Vincent Feuillard & Anne Sabourin, 2020. "A multivariate extreme value theory approach to anomaly clustering and visualization," Computational Statistics, Springer, vol. 35(2), pages 607-628, June.
    2. Sebastian Engelke & Adrien S. Hitz, 2020. "Graphical models for extremes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 82(4), pages 871-932, September.
    3. Rootzen, Holger & Segers, Johan & Wadsworth, Jennifer, 2018. "Multivariate peaks over thresholds models," LIDAM Reprints ISBA 2018005, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    4. Einmahl, J.H.J. & Krajina, A. & Segers, J., 2011. "An M-Estimator for Tail Dependence in Arbitrary Dimensions," Discussion Paper 2011-013, Tilburg University, Center for Economic Research.
    5. Rootzen, Holger & Segers, Johan & Wadsworth, Jennifer L., 2018. "Multivariate generalized Pareto distributions: Parametrizations, representations, and properties," LIDAM Reprints ISBA 2018003, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    6. Clément Dombry & Sebastian Engelke & Marco Oesting, 2016. "Exact simulation of max-stable processes," Biometrika, Biometrika Trust, vol. 103(2), pages 303-317.
    7. Ho, Zhen Wai Olivier & Dombry, Clément, 2019. "Simple models for multivariate regular variation and the Hüsler–Reiß Pareto distribution," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 525-550.
    8. Goix, Nicolas & Sabourin, Anne & Clémençon, Stephan, 2017. "Sparse representation of multivariate extremes with applications to anomaly detection," Journal of Multivariate Analysis, Elsevier, vol. 161(C), pages 12-31.
    9. Chiapino, Mael & Sabourin, Anne & Segers, Johan, 2019. "Identifying groups of variables with the potential of being large simultaneously," LIDAM Reprints ISBA 2019021, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    10. Cooley, Daniel & Davis, Richard A. & Naveau, Philippe, 2010. "The pairwise beta distribution: A flexible parametric multivariate model for extremes," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 2103-2117, October.
    11. F. Ballani & M. Schlather, 2011. "A construction principle for multivariate extreme value distributions," Biometrika, Biometrika Trust, vol. 98(3), pages 633-645.
    12. V Fomichov & J Ivanovs, 2023. "Spherical clustering in detection of groups of concomitant extremes," Biometrika, Biometrika Trust, vol. 110(1), pages 135-153.
    13. Hüsler, Jürg & Reiss, Rolf-Dieter, 1989. "Maxima of normal random vectors: Between independence and complete dependence," Statistics & Probability Letters, Elsevier, vol. 7(4), pages 283-286, February.
    14. Einmahl, J.H.J. & Krajina, A. & Segers, J., 2011. "An M-Estimator for Tail Dependence in Arbitrary Dimensions," Discussion Paper 2011-013, Tilburg University, Center for Economic Research.
    15. Drees, Holger & Huang, Xin, 1998. "Best Attainable Rates of Convergence for Estimators of the Stable Tail Dependence Function," Journal of Multivariate Analysis, Elsevier, vol. 64(1), pages 25-47, January.
    Full references (including those not matched with items on IDEAS)

    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. Hentschel, Manuel & Engelke, Sebastian & Segers, Johan, 2022. "Statistical Inference for Hüsler–Reiss Graphical Models Through Matrix Completions," LIDAM Discussion Papers ISBA 2022032, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    2. Hu, Shuang & Peng, Zuoxiang & Segers, Johan, 2022. "Modelling multivariate extreme value distributions via Markov trees," LIDAM Discussion Papers ISBA 2022021, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    3. Einmahl, John & Segers, Johan, 2020. "Empirical Tail Copulas for Functional Data," Discussion Paper 2020-004, Tilburg University, Center for Economic Research.
    4. Segers, Johan, 2012. "Max-Stable Models For Multivariate Extremes," LIDAM Discussion Papers ISBA 2012011, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    5. Einmahl, John & Segers, Johan, 2020. "Empirical Tail Copulas for Functional Data," Discussion Paper 2020-004, Tilburg University, Center for Economic Research.
    6. Asenova, Stefka & Segers, Johan, 2022. "Extremes of Markov random fields on block graphs," LIDAM Discussion Papers ISBA 2022013, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    7. Asenova, Stefka Kirilova & Mazo, Gildas & Segers, Johan, 2020. "Inference on extremal dependence in a latent Markov tree model attracted to a Husler-Reiss distribution," LIDAM Discussion Papers ISBA 2020005, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    8. Bücher, Axel & Jäschke, Stefan & Wied, Dominik, 2015. "Nonparametric tests for constant tail dependence with an application to energy and finance," Journal of Econometrics, Elsevier, vol. 187(1), pages 154-168.
    9. Goix, Nicolas & Sabourin, Anne & Clémençon, Stephan, 2017. "Sparse representation of multivariate extremes with applications to anomaly detection," Journal of Multivariate Analysis, Elsevier, vol. 161(C), pages 12-31.
    10. Gissibl, Nadine & Klüppelberg, Claudia & Otto, Moritz, 2018. "Tail dependence of recursive max-linear models with regularly varying noise variables," Econometrics and Statistics, Elsevier, vol. 6(C), pages 149-167.
    11. Bücher, Axel & Volgushev, Stanislav & Zou, Nan, 2019. "On second order conditions in the multivariate block maxima and peak over threshold method," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 604-619.
    12. John H. J. Einmahl & Anna Kiriliouk & Andrea Krajina & Johan Segers, 2016. "An M-estimator of spatial tail dependence," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(1), pages 275-298, January.
    13. Kiriliouk, Anna, 2020. "Hypothesis testing for tail dependence parameters on the boundary of the parameter space," Econometrics and Statistics, Elsevier, vol. 16(C), pages 121-135.
    14. Einmahl, John & Kiriliouk, Anna & Segers, Johan, 2016. "A continuous updating weighted least squares estimator of tail dependence in high dimensions," LIDAM Discussion Papers ISBA 2016002, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    15. 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).
    16. Simpson, Emma S. & Wadsworth, Jennifer L. & Tawn, Jonathan A., 2021. "A geometric investigation into the tail dependence of vine copulas," Journal of Multivariate Analysis, Elsevier, vol. 184(C).
    17. 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).
    18. 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).
    19. Asenova, Stefka & Segers, Johan, 2022. "Max-linear graphical models with heavy-tailed factors on trees of transitive tournaments," LIDAM Discussion Papers ISBA 2022031, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    20. Klüppelberg, Claudia & Krali, Mario, 2021. "Estimating an extreme Bayesian network via scalings," Journal of Multivariate Analysis, Elsevier, vol. 181(C).

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    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:aiz:louvad:2023034. 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: Nadja Peiffer (email available below). General contact details of provider: https://edirc.repec.org/data/isuclbe.html .

    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.