Skip to main content
SpringerLink
Log in

Multivariate extreme value copulas with factor and tree dependence structures

  • Published:
Extremes Aims and scope Submit manuscript

Abstract

Parsimonious extreme value copula models with O(d) parameters for d observed variables of extrema are presented. These models utilize the dependence characteristics, including factor and tree structures, assumed on the underlying variables that give rise to the data of extremes. For factor structures, a class of parametric models is obtained by taking the extreme value limit of factor copulas with non-zero tail dependence. An alternative model suitable for both factor and tree structures imposes constraints on the parametric Hüsler-Reiss copula to get representations in terms of O(d) other parameters. Dependence properties are discussed. As the full density is often intractable, the method of composite (pairwise) likelihood is used for model inference. Procedures to improve the stability of bivariate density evaluation are also developed. The proposed models are applied to two data examples — one for annual extreme river flows and one for bimonthly extremes of daily stock returns.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Aas, K., Czado, C., Frigessi, A., Bakken, H.: Pair-copula constructions of multiple dependence. Insurance: Math. Econ. 44, 182–198 (2009)

    MathSciNet  MATH  Google Scholar 

  • Bedford, T., Cooke, R.M.: Probability density decomposition for conditionally dependent random variables modeled by vines. Ann. Math. Artif. Intell. 32, 245–268 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Bedford, T., Cooke, R.M.: Vines — a new graphical model for dependent random variables. Ann. Stat. 30, 1031–1068 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Besag, J.: Spatial interaction and the statistical analysis of lattice systems. J. R. Stat. Soc. Ser. B 36, 192–236 (1974)

    MathSciNet  MATH  Google Scholar 

  • Brechmann, E.C., Czado, C., Aas, K.: Truncated regular vines in high dimensions with application to financial data. Can. J. Stat. 40, 68–85 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Burr, I.W.: Cumulative frequency functions. Ann. Math. Stat. 13, 215–232 (1942)

    Article  MathSciNet  MATH  Google Scholar 

  • Canadian Heritage Rivers System: Fraser River. http://chrs.ca/the-rivers/fraser/. Accessed on August 14, 2016 (2016)

  • Carlson, M.: A Brief History of the 1987 Stock Market Crash with a Discussion of the Federal Reserve Response. Finance and Economics Discussion Series, Federal Reserve Board of Governors (2007)

  • Clayton, D.G.: A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65, 141–151 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  • Coles, S.: An Introduction to Statistical Modeling of Extreme Values. Springer, London (2001)

    Book  MATH  Google Scholar 

  • Cook, R.D., Johnson, M.E.: A family of distributions for modelling non-elliptically symmetric multivariate data. J. R. Stat. Soc. Ser. B 43, 210–218 (1981)

    MATH  Google Scholar 

  • Cox, D.R., Reid, N.: A note on pseudolikelihood constructed from marginal densities. Biometrika 91, 729–737 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Dagum, C.: A model of income distribution and the conditions of existence of moments of finite order Proceedings of the 40th Session of the International Statistical Institute, vol. 46, pp 199–205 (1975)

  • Davison, A.C., Padoan, S.A., Ribatet, M.: Statistical modeling of spatial extremes. Stat. Sci. 27, 161–186 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Demarta, S., McNeil, A.J.: The t copula and related copulas. Int. Stat. Rev. 73, 111–129 (2005)

    Article  MATH  Google Scholar 

  • Drees, H., Huang, X.: Best attainable rates of convergence for estimators of the stable tail dependence function. J. Multivar. Anal. 64, 25–46 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Galambos, J.: Order statistics of samples from multivariate distributions. J. Am. Stat. Assoc. 70, 674–680 (1975)

    MathSciNet  MATH  Google Scholar 

  • Galambos, J. The Asymptotic Theory of Extreme Order Statistics: , 2nd edn. Krieger Publishing Co., Malabar (1987)

  • Gao, X., Song, P.X.K.: Composite likelihood Bayesian information criteria for model selection in high-dimensional data. J. Am. Stat. Assoc. 105, 1531–1540 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Genton, M.G., Ma, Y., Sang, H.: On the likelihood function of Gaussian max-stable processes. Biometrika 98, 481–488 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Godambe, V.P.: An optimum property of regular maximum likelihood estimation. Ann. Math. Stat. 31, 1208–1211 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  • Gumbel, E.J.: Distributions des valeurs extrêmes en plusieurs dimensions. Publ. de l’Institut de Statistique de l’Université, de Paris 9, 171–173 (1960)

    MATH  Google Scholar 

  • Huang, X.: Statistics of bivariate extremes. PhD thesis, Erasmus University Rotterdam, Tinbergen Institute Research Series 22 (1992)

  • Huser, R., Davison, A.C.: Composite likelihood estimation for the Brown-Resnick process. Biometrika 100, 511–518 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Hüsler, J., Reiss, R.D.: Maxima of normal random vectors: Between independence and complete dependence. Stat. Probab. Lett. 7, 283–286 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  • Joe, H.: Parametric families of multivariate distributions with given margins. J. Multivar. Anal. 46, 262–282 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  • Joe, H.: Dependence Modeling with Copulas. Chapman & Hall, Boca Raton (2014)

    MATH  Google Scholar 

  • Joe, H., Li, H., Nikoloulopoulos, A.K.: Tail dependence functions and vine copulas. J. Multivar. Anal. 101, 252–270 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Krupskii, P., Joe, H.: Factor copula models for multivariate data. J. Multivar. Anal. 120, 85–101 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Lindsay, B.G.: Composite likelihood methods. Contemp. Math. 80, 221–239 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  • Mardia, K.V.: Multivariate Pareto distributions. Ann. Math. Stat. 33, 1008–1015 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  • Nikoloulopoulos, A.K., Joe, H., Li, H.: Extreme value properties of multivariate t copulas. Extremes 12, 129–148 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Padoan, S.A.: Multivariate extreme models based on underlying skew-t and skew-normal distributions. J. Multivar. Anal. 102, 977–991 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Padoan, S.A., Ribatet, M., Sisson, S.A.: Likelihood-based inference for max-stable processes. J. Am. Stat. Assoc. 105, 263–277 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Pickands, J.: Multivariate extreme value distributions Proceedings 43rd Session International Statistical Institute, vol. 2, pp 859–878 (1981)

  • Ribatet, M.: Spatial extremes: Max-stable processes at work. J de la Socié,té Franċaise de Statistique 154, 156–177 (2013)

    MathSciNet  MATH  Google Scholar 

  • Schmidt, R., Stadtmüller, U.: Non-parametric estimation of tail dependence. Scand. J. Stat. 33, 307–335 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Singh, S.K., Maddala, G.S.: A function for size distribution of incomes. Econometrica 44, 963–970 (1976)

    Article  MATH  Google Scholar 

  • Sklar, A.: Fonctions de répartition à n dimensions et leurs marges. Publications de l’Institut de Statistique de l’Université, de Paris 8, 229–231 (1959)

    MATH  Google Scholar 

  • Smith, R.L.: Max-stable Processes and Spatial Extremes. Unpublished Manuscript (1990)

  • Smith, R.L.: Measuring risk with extreme value theory. In: Dempster, M.A.H. (ed.) Risk Management: Value at Risk and Beyond, pp 224–246. Cambridge University Press (2002)

  • Stroud, A., Secrest, D.: Gaussian Quadrature Formulas Prentice-Hall. Englewood Cliffs, NJ (1966)

    MATH  Google Scholar 

  • Takahasi, K.: Note on the multivariate Burr’s distribution. Ann. Inst. Stat. Math. 17, 257–260 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  • Tsay, R.S. Analysis of Financial Time Series: , 3rd edn. Wiley, Hoboken (2010)

  • Varin, C., Vidoni, P.: A note on composite likelihood inference and model selection. Biometrika 92, 519–528 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Varin, C., Reid, N., Firth, D.: An overview of composite likelihood methods. Stat. Sin. 21, 5–42 (2011)

    MathSciNet  MATH  Google Scholar 

  • Zhao, Y., Joe, H.: Composite likelihood estimation in multivariate data analysis. Can. J. Stat. 33, 335–356 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This research has been supported by UBC’s Four Year Doctoral Fellowship and NSERC Discovery Grant 8698. We would like to thank the associate editor and the anonymous referees for their helpful comments, which improved the presentation of the paper.

Author information

Authors and Affiliations

  1. Department of Statistics, University of British Columbia, Vancouver, BC, V6T 1Z4, Canada

    David Lee & Harry Joe

Authors
  1. David Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Harry Joe
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David Lee.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lee, D., Joe, H. Multivariate extreme value copulas with factor and tree dependence structures. Extremes 21, 147–176 (2018). https://doi.org/10.1007/s10687-017-0298-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10687-017-0298-0

Keywords

AMS 2000 Subject Classifications

Search

Navigation