IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v122y2013icp409-413.html
   My bibliography  Save this article

Extremal t processes: Elliptical domain of attraction and a spectral representation

Author

Listed:
  • Opitz, T.

Abstract

The extremal t process was proposed in the literature for modeling spatial extremes within a copula framework based on the extreme value limit of elliptical t distributions (Davison et al. (2012) [5]). A major drawback of this max-stable model was the lack of a spectral representation such that for instance direct simulation was infeasible. The main contribution of this note is to propose such a spectral construction for the extremal t process. Interestingly, the extremal Gaussian process introduced by Schlather (2002) [22] appears as a special case. We further highlight the role of the extremal t process as the maximum attractor for processes with finite-dimensional elliptical distributions. All results naturally also hold within the multivariate domain.

Suggested Citation

  • Opitz, T., 2013. "Extremal t processes: Elliptical domain of attraction and a spectral representation," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 409-413.
    • Handle: RePEc:eee:jmvana:v:122:y:2013:i:c:p:409-413
    DOI: 10.1016/j.jmva.2013.08.008
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047259X13001693
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.jmva.2013.08.008?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. Davis, Richard A. & Mikosch, Thomas, 2008. "Extreme value theory for space-time processes with heavy-tailed distributions," Stochastic Processes and their Applications, Elsevier, vol. 118(4), pages 560-584, April.
    2. Segers, Johan, 2012. "Max-stable models for multivariate extremes," LIDAM Reprints ISBA 2012012, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    3. Hashorva, Enkelejd, 2005. "Elliptical triangular arrays in the max-domain of attraction of Hüsler-Reiss distribution," Statistics & Probability Letters, Elsevier, vol. 72(2), pages 125-135, April.
    4. C. Dombry & F. Éyi-Minko & M. Ribatet, 2013. "Conditional simulation of max-stable processes," Biometrika, Biometrika Trust, vol. 100(1), pages 111-124.
    5. 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).
    6. Hashorva, Enkelejd, 2006. "On the regular variation of elliptical random vectors," Statistics & Probability Letters, Elsevier, vol. 76(14), pages 1427-1434, August.
    7. 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.
    8. Martin Schlather, 2003. "A dependence measure for multivariate and spatial extreme values: Properties and inference," Biometrika, Biometrika Trust, vol. 90(1), pages 139-156, March.
    9. Claudia Klüppelberg & Gabriel Kuhn & Liang Peng, 2008. "Semi‐Parametric Models for the Multivariate Tail Dependence Function – the Asymptotically Dependent Case," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 35(4), pages 701-718, December.
    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. Hofert, Marius & Huser, Raphaël & Prasad, Avinash, 2018. "Hierarchical Archimax copulas," Journal of Multivariate Analysis, Elsevier, vol. 167(C), pages 195-211.
    2. Koch, Erwan & Robert, Christian Y., 2022. "Stochastic derivative estimation for max-stable random fields," European Journal of Operational Research, Elsevier, vol. 302(2), pages 575-588.
    3. Zhong, Peng & Huser, Raphaël & Opitz, Thomas, 2024. "Exact Simulation of Max-Infinitely Divisible Processes," Econometrics and Statistics, Elsevier, vol. 30(C), pages 96-109.
    4. Hashorva, Enkelejd, 2018. "Representations of max-stable processes via exponential tilting," Stochastic Processes and their Applications, Elsevier, vol. 128(9), pages 2952-2978.
    5. Papastathopoulos, Ioannis & Tawn, Jonathan A., 2016. "Conditioned limit laws for inverted max-stable processes," Journal of Multivariate Analysis, Elsevier, vol. 150(C), pages 214-228.
    6. Raphaël Huser & Marc G. Genton, 2016. "Non-Stationary Dependence Structures for Spatial Extremes," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 21(3), pages 470-491, September.
    7. Lee, Xing Ju & Hainy, Markus & McKeone, James P. & Drovandi, Christopher C. & Pettitt, Anthony N., 2018. "ABC model selection for spatial extremes models applied to South Australian maximum temperature data," Computational Statistics & Data Analysis, Elsevier, vol. 128(C), pages 128-144.
    8. Padoan, Simone A. & Bevilacqua, Moreno, 2015. "Analysis of Random Fields Using CompRandFld," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 63(i09).
    9. Harald Schellander & Tobias Hell, 2018. "Modeling snow depth extremes in Austria," 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. 94(3), pages 1367-1389, December.
    10. Belzile, Léo R. & Nešlehová, Johanna G., 2017. "Extremal attractors of Liouville copulas," Journal of Multivariate Analysis, Elsevier, vol. 160(C), pages 68-92.

    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. Jäschke, Stefan, 2014. "Estimation of risk measures in energy portfolios using modern copula techniques," Computational Statistics & Data Analysis, Elsevier, vol. 76(C), pages 359-376.
    2. 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.
    3. Whitney K. Huang & Daniel S. Cooley & Imme Ebert-Uphoff & Chen Chen & Snigdhansu Chatterjee, 2019. "New Exploratory Tools for Extremal Dependence: $$\chi $$ χ Networks and Annual Extremal Networks," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 24(3), pages 484-501, September.
    4. Hashorva, Enkelejd & Weng, Zhichao, 2013. "Limit laws for extremes of dependent stationary Gaussian arrays," Statistics & Probability Letters, Elsevier, vol. 83(1), pages 320-330.
    5. Krupskii, Pavel & Huser, Raphaël, 2024. "Max-convolution processes with random shape indicator kernels," Journal of Multivariate Analysis, Elsevier, vol. 203(C).
    6. Pavel Krupskii, 2017. "Copula-based measures of reflection and permutation asymmetry and statistical tests," Statistical Papers, Springer, vol. 58(4), pages 1165-1187, December.
    7. Frick, Melanie & Reiss, Rolf-Dieter, 2013. "Expansions and penultimate distributions of maxima of bivariate normal random vectors," Statistics & Probability Letters, Elsevier, vol. 83(11), pages 2563-2568.
    8. 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.
    9. 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).
    10. Carsten Bormann & Julia Schaumburg & Melanie Schienle, 2016. "Beyond Dimension two: A Test for Higher-Order Tail Risk," Journal of Financial Econometrics, Oxford University Press, vol. 14(3), pages 552-580.
    11. 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.
    12. Kabluchko, Zakhar, 2009. "Extremes of space-time Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 119(11), pages 3962-3980, November.
    13. Padoan, Simone A., 2011. "Multivariate extreme models based on underlying skew-t and skew-normal distributions," Journal of Multivariate Analysis, Elsevier, vol. 102(5), pages 977-991, May.
    14. 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.
    15. Koch, Erwan & Robert, Christian Y., 2022. "Stochastic derivative estimation for max-stable random fields," European Journal of Operational Research, Elsevier, vol. 302(2), pages 575-588.
    16. Belzile, Léo R. & Nešlehová, Johanna G., 2017. "Extremal attractors of Liouville copulas," Journal of Multivariate Analysis, Elsevier, vol. 160(C), pages 68-92.
    17. Lee, Xing Ju & Hainy, Markus & McKeone, James P. & Drovandi, Christopher C. & Pettitt, Anthony N., 2018. "ABC model selection for spatial extremes models applied to South Australian maximum temperature data," Computational Statistics & Data Analysis, Elsevier, vol. 128(C), pages 128-144.
    18. Krupskii, Pavel & Genton, Marc G., 2019. "A copula model for non-Gaussian multivariate spatial data," Journal of Multivariate Analysis, Elsevier, vol. 169(C), pages 264-277.
    19. Manjunath, B.G. & Frick, Melanie & Reiss, Rolf-Dieter, 2012. "Some notes on extremal discriminant analysis," Journal of Multivariate Analysis, Elsevier, vol. 103(1), pages 107-115, January.
    20. Kiriliouk, Anna & Segers, Johan & Warchol, Michal, 2014. "Nonparametric estimation of extremal dependence," LIDAM Discussion Papers ISBA 2014044, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).

    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:122:y:2013:i:c:p:409-413. 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.