Electronic Journal of Statistics

Bayesian inference for the extremal dependence

Giulia Marcon, Simone A. Padoan, and Isadora Antoniano-Villalobos

Full-text: Open access

Abstract

A simple approach for modeling multivariate extremes is to consider the vector of component-wise maxima and their max-stable distributions. The extremal dependence can be inferred by estimating the angular measure or, alternatively, the Pickands dependence function. We propose a nonparametric Bayesian model that allows, in the bivariate case, the simultaneous estimation of both functional representations through the use of polynomials in the Bernstein form. The constraints required to provide a valid extremal dependence are addressed in a straightforward manner, by placing a prior on the coefficients of the Bernstein polynomials which gives probability one to the set of valid functions. The prior is extended to the polynomial degree, making our approach nonparametric. Although the analytical expression of the posterior is unknown, inference is possible via a trans-dimensional MCMC scheme. We show the efficiency of the proposed methodology by means of a simulation study. The extremal behaviour of log-returns of daily exchange rates between the Pound Sterling vs the U.S. Dollar and the Pound Sterling vs the Japanese Yen is analysed for illustrative purposes.

Article information

Source
Electron. J. Statist., Volume 10, Number 2 (2016), 3310-3337.

Dates
Received: December 2015
First available in Project Euclid: 16 November 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1479287223

Digital Object Identifier
doi:10.1214/16-EJS1162

Mathematical Reviews number (MathSciNet)
MR3572851

Zentralblatt MATH identifier
1357.62213

Subjects
Primary: 62G05: Estimation 62G07: Density estimation 62G32: Statistics of extreme values; tail inference

Keywords
Generalised extreme value distribution extremal dependence angular measure max-stable distribution Bernstein polynomials Bayesian nonparametrics trans-dimensional MCMC exchange rates

Citation

Marcon, Giulia; Padoan, Simone A.; Antoniano-Villalobos, Isadora. Bayesian inference for the extremal dependence. Electron. J. Statist. 10 (2016), no. 2, 3310--3337. doi:10.1214/16-EJS1162. https://projecteuclid.org/euclid.ejs/1479287223


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References

  • Antoniano-Villalobos, I. and S. G. Walker (2013). Bayesian nonparametric inference for the power likelihood., Journal of Computational and Graphical Statistics 22(4), 801–813.
  • Beirlant, J., Y. Goegebeur, J. Segers, and J. Teugels (2004)., Statistics of Extremes: Theory and Applications. John Wiley & Sons Ltd., Chichester.
  • Beranger, B. and S. A. Padoan (2015). Extreme dependence models. In D. Dey and J. Yan (Eds.), Extreme Value Modeling and Risk Analysis: Methods and Applications. Chapman and Hall/CRC.
  • Berghaus, B., A. Bücher, and H. Dette (2013). Minimum distance estimators of the Pickands dependence function and related tests of multivariate extreme-value dependence., Journal de la Société Française de Statistique 154(1), 116–137.
  • Boldi, M. O. and A. C. Davison (2007). A mixture model for multivariate extremes., Journal of the Royal Statistical Society, Series B 69(2), 217–229.
  • Bücher, A., H. Dette, and S. Volgushev (2011). New estimators of the Pickands dependence function and a test for extreme-value dependence., The Annals of Statistics 39(4), 1963–2006.
  • Capéraà, P., A.-L. Fougères, and C. Genest (1997). A nonparametric estimation procedure for bivariate extreme value copulas., Biometrika 84, 567–577.
  • Coles, S. G. (2001)., An Introduction to Statistical Modelling of Extreme Values. Springer, London.
  • de Haan, L. and A. Ferreira (2006)., Extreme Value Theory: An Introduction. Springer.
  • Einmahl, J., A. Krajina, and J. Segers (2008). A method of moments estimator of tail dependence., Bernoulli 14, 1003–1026.
  • Engel, C. and K. D. West (2005). Exchange rates and fundamentals., Journal of Political Economy 113(3), 485–517.
  • Falk, M., J. Hüsler, and R. D. Reiss (2010)., Laws of Small Numbers: Extremes and Rare Events (Third ed.). Birkhäuser Boston.
  • Fils-Villetard, A., A. Guillou, and J. Segers (2008). Projection estimators of Pickands dependence functions., The Canadian Journal of Statistics 36(3), 369–382.
  • Genest, C. and J. Segers (2009). Rank-based inference for bivariate extreme-value copulas., The Annals of Statistics 37(5B), 2990–3022.
  • Ghosal, S. (2001). Convergence rates for density estimation with Bernstein polynomials., The Annals of Statistics 29(5), 1264–1280.
  • Godsill, S. J. (2001). On the relationship between Markov Chain Monte Carlo methods for model uncertainty., Journal of Computational and Graphical Statistics 10(2), 230–248.
  • Guillotte, S. and F. Perron (2008). A Bayesian estimator for the dependence function of a bivariate extreme-value distribution., Canadian Journal of Statistics 36(3), 383–396.
  • Hüsler, J. and R. Reiss (1989). Maxima of normal random vectors: between independence and complete dependence., Statistics and Probability Letters 7, 283–286.
  • Klüppelberg, C., G. Kuhn, L. Peng, et al. (2007). Estimating the tail dependence function of an elliptical distribution., Bernoulli 13(1), 229–251.
  • Krajina, A. (2012). A method of moments estimator of tail dependence in meta-elliptical models., Journal of Statistical Planning and Inference 142(7), 1811–1823.
  • Lorentz, G. G. (1986)., Bernstein Polynominals (Second ed.). Chelsea Publishing Company, New York.
  • Madura, J. (2014)., Financial markets and institutions. Cengage learning.
  • Marcon, G., S. A. Padoan, P. Naveau, and P. Muliere (2015). Nonparametric estimation of the Pickands dependence function using Bernstein polynomials., Journal of Statistical Planning and Inference, Under revision.
  • Meese, R. A. and K. Rogoff (1983). Empirical exchange rate models of the seventies: Do they fit out of sample?, Journal of International Economics 14(1), 3–24.
  • Nikoloulopoulos, A. K., H. Joe, and H. Li (2009). Extreme value properties of multivariate t copulas., Extremes 12(2), 129–148.
  • Petrone, S. (1999a). Bayesian density estimation using Bernstein polynomials., Canadian Journal of Statistics 27(1), 105–126.
  • Petrone, S. (1999b). Random Bernstein polynomials., Scandinavian Journal of Statistics 26, 373–393.
  • Petrone, S. and L. Wasserman (2002). Consistency of Bernstein polynomial posteriors., Journal of the Royal Statistical Society: Series B (Statistical Methodology) 64(1), 79–100.
  • Pickands, III, J. (1981). Multivariate extreme value distributions. In, Proceedings of the 43rd session of the International Statistical Institute, Vol. 2 (Buenos Aires, 1981), Volume 49, pp. 859–878, 894–902. With a discussion.
  • Rockafellar, R. T. (2015)., Convex analysis. Princeton university press.
  • Sabourin, A. and P. Naveau (2014). Bayesian Dirichlet mixture model for multivariate extremes: A re-parametrization., Computational Statistics & Data Analysis 71, 542–567.
  • Stephenson, A. (2004). A user’s guide to the ‘evd’ package (version 2.1)., Department of Statistics. Macquarie University. Australia.
  • Tawn, J. A. (1990). Modelling multivariate extreme value distributions., Biometrika 77(2), 245–253.