IDEAS home Printed from https://ideas.repec.org/a/eee/csdana/v173y2022ics016794732200086x.html
   My bibliography  Save this article

Stochastic representation of FGM copulas using multivariate Bernoulli random variables

Author

Listed:
  • Blier-Wong, Christopher
  • Cossette, Hélène
  • Marceau, Etienne

Abstract

A one-to-one correspondence between Fréchet's class of multivariate Bernoulli distribution with symmetric marginals and the well-known family of Farlie-Gumbel-Morgenstern (FGM) copulas is established. A new stochastic representation of the family of d-variate FGM copulas is introduced. The representation is bijective: from any d-variate Bernoulli distribution, one may define a corresponding d-variate FGM copula; and for any d-variate FGM copula, one finds the corresponding d-variate Bernoulli distribution. The proposed stochastic representation has many advantages, notably establishing stochastic orders, constructing subclasses of FGM copulas and sampling. In particular, one may use the stochastic representation to develop computational methods to perform sampling from subclasses of FGM copulas, which scale well to large dimensions.

Suggested Citation

  • Blier-Wong, Christopher & Cossette, Hélène & Marceau, Etienne, 2022. "Stochastic representation of FGM copulas using multivariate Bernoulli random variables," Computational Statistics & Data Analysis, Elsevier, vol. 173(C).
    • Handle: RePEc:eee:csdana:v:173:y:2022:i:c:s016794732200086x
    DOI: 10.1016/j.csda.2022.107506
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S016794732200086X
    Download Restriction: Full text for ScienceDirect subscribers only.
    File URL: https://libkey.io/10.1016/j.csda.2022.107506?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. David M. Blei & Alp Kucukelbir & Jon D. McAuliffe, 2017. "Variational Inference: A Review for Statisticians," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(518), pages 859-877, April.
    2. Baker, Rose, 2008. "An order-statistics-based method for constructing multivariate distributions with fixed marginals," Journal of Multivariate Analysis, Elsevier, vol. 99(10), pages 2312-2327, November.
    3. Berghaus, Betina & Segers, Johan, 2017. "Weak convergence of the weighted empirical beta copula process," LIDAM Discussion Papers ISBA 2017015, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    4. Joe, Harry, 1990. "Multivariate concordance," Journal of Multivariate Analysis, Elsevier, vol. 35(1), pages 12-30, October.
    5. Teugels, Jozef L, 1990. "Some representations of the multivariate Bernoulli and binomial distributions," Journal of Multivariate Analysis, Elsevier, vol. 32(2), pages 256-268, February.
    6. Wei Jiang & Shuang Song & Lin Hou & Hongyu Zhao, 2021. "A Set of Efficient Methods to Generate High-Dimensional Binary Data With Specified Correlation Structures," The American Statistician, Taylor & Francis Journals, vol. 75(3), pages 310-322, July.
    7. Segers, Johan & Sibuya, Masaaki & Tsukahara, Hideatsu, 2017. "The empirical beta copula," LIDAM Reprints ISBA 2017005, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    8. Gijbels, Irène & Kika, Vojtěch & Omelka, Marek, 2021. "On the specification of multivariate association measures and their behaviour with increasing dimension," Journal of Multivariate Analysis, Elsevier, vol. 182(C).
    9. Schmid, Friedrich & Schmidt, Rafael, 2007. "Multivariate extensions of Spearman's rho and related statistics," Statistics & Probability Letters, Elsevier, vol. 77(4), pages 407-416, February.
    10. Segers, Johan & Sibuya, Masaaki & Tsukahara, Hideatsu, 2017. "The empirical beta copula," Journal of Multivariate Analysis, Elsevier, vol. 155(C), pages 35-51.
    11. César García‐Gómez & Ana Pérez & Mercedes Prieto‐Alaiz, 2021. "Copula‐based analysis of multivariate dependence patterns between dimensions of poverty in Europe," Review of Income and Wealth, International Association for Research in Income and Wealth, vol. 67(1), pages 165-195, March.
    12. Cambanis, Stamatis, 1977. "Some properties and generalizations of multivariate Eyraud-Gumbel-Morgenstern distributions," Journal of Multivariate Analysis, Elsevier, vol. 7(4), pages 551-559, December.
    13. Alexander J. McNeil & Rüdiger Frey & Paul Embrechts, 2015. "Quantitative Risk Management: Concepts, Techniques and Tools Revised edition," Economics Books, Princeton University Press, edition 2, number 10496.
    14. Bargès, Mathieu & Cossette, Hélène & Loisel, Stéphane & Marceau, Étienne, 2011. "On the Moments of Aggregate Discounted Claims with Dependence Introduced by a FGM Copula," ASTIN Bulletin, Cambridge University Press, vol. 41(1), pages 215-238, May.
    15. Fontana, Roberto & Semeraro, Patrizia, 2018. "Representation of multivariate Bernoulli distributions with a given set of specified moments," Journal of Multivariate Analysis, Elsevier, vol. 168(C), pages 290-303.
    16. Sharakhmetov, Sh. & Ibragimov, R., 2002. "A Characterization of Joint Distribution of Two-Valued Random Variables and Its Applications," Journal of Multivariate Analysis, Elsevier, vol. 83(2), pages 389-408, November.
    17. Pérez, Ana & Prieto-Alaiz, Mercedes, 2016. "A note on nonparametric estimation of copula-based multivariate extensions of Spearman’s rho," Statistics & Probability Letters, Elsevier, vol. 112(C), pages 41-50.
    18. Cossette, Hélène & Marceau, Etienne & Trufin, Julien & Zuyderhoff, Pierre, 2020. "Ruin-based risk measures in discrete-time risk models," Insurance: Mathematics and Economics, Elsevier, vol. 93(C), pages 246-261.
    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. Blier-Wong, Christopher & Cossette, Hélène & Marceau, Etienne, 2023. "Risk aggregation with FGM copulas," Insurance: Mathematics and Economics, Elsevier, vol. 111(C), pages 102-120.

    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. Aleksy Leeuwenkamp & Wentao Hu, 2023. "New general dependence measures: construction, estimation and application to high-frequency stock returns," Papers 2309.00025, arXiv.org.
    2. Hofert, Marius & Prasad, Avinash & Zhu, Mu, 2022. "Multivariate time-series modeling with generative neural networks," Econometrics and Statistics, Elsevier, vol. 23(C), pages 147-164.
    3. Lu Lu & Sujit Ghosh, 2023. "Nonparametric Estimation of Multivariate Copula Using Empirical Bayes Methods," Mathematics, MDPI, vol. 11(20), pages 1-22, October.
    4. Eddie Anderson & Artem Prokhorov & Yajing Zhu, 2020. "A Simple Estimator of Two‐Dimensional Copulas, with Applications," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 82(6), pages 1375-1412, December.
    5. Blier-Wong, Christopher & Cossette, Hélène & Marceau, Etienne, 2023. "Risk aggregation with FGM copulas," Insurance: Mathematics and Economics, Elsevier, vol. 111(C), pages 102-120.
    6. Junker, Robert R. & Griessenberger, Florian & Trutschnig, Wolfgang, 2021. "Estimating scale-invariant directed dependence of bivariate distributions," Computational Statistics & Data Analysis, Elsevier, vol. 153(C).
    7. Shih, Jia-Han & Emura, Takeshi, 2021. "On the copula correlation ratio and its generalization," Journal of Multivariate Analysis, Elsevier, vol. 182(C).
    8. Dietmar Pfeifer & Olena Ragulina, 2020. "Adaptive Bernstein Copulas and Risk Management," Papers 2011.00909, arXiv.org, revised Mar 2021.
    9. Dietmar Pfeifer & Olena Ragulina, 2020. "Adaptive Bernstein Copulas and Risk Management," Mathematics, MDPI, vol. 8(12), pages 1-22, December.
    10. Laverny, Oskar & Masiello, Esterina & Maume-Deschamps, Véronique & Rullière, Didier, 2021. "Dependence structure estimation using Copula Recursive Trees," Journal of Multivariate Analysis, Elsevier, vol. 185(C).
    11. 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.
    12. 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).
    13. Gijbels, Irène & Kika, Vojtěch & Omelka, Marek, 2021. "On the specification of multivariate association measures and their behaviour with increasing dimension," Journal of Multivariate Analysis, Elsevier, vol. 182(C).
    14. Kojadinovic, Ivan & Stemikovskaya, Kristina, 2019. "Subsampling (weighted smooth) empirical copula processes," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 704-723.
    15. 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).
    16. Kiriliouk, Anna & Segers, Johan & Tsukahara, Hideatsu, 2019. "On Some Resampling Procedures with the Empirical Beta Copula," LIDAM Discussion Papers ISBA 2019012, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    17. Berghaus, Betina & Segers, Johan, 2017. "Weak convergence of the weighted empirical beta copula process," LIDAM Discussion Papers ISBA 2017015, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    18. Grothe, Oliver & Schnieders, Julius & Segers, Johan, 2013. "Measuring Association and Dependence Between Random Vectors," LIDAM Discussion Papers ISBA 2013026, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    19. Pérez, Ana & Prieto-Alaiz, Mercedes, 2016. "A note on nonparametric estimation of copula-based multivariate extensions of Spearman’s rho," Statistics & Probability Letters, Elsevier, vol. 112(C), pages 41-50.
    20. 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.

    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:csdana:v:173:y:2022:i:c:s016794732200086x. 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/locate/csda .

    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.