The Annals of Statistics

An M-estimator for tail dependence in arbitrary dimensions

John H. J. Einmahl, Andrea Krajina, and Johan Segers

Full-text: Open access

Abstract

Consider a random sample in the max-domain of attraction of a multivariate extreme value distribution such that the dependence structure of the attractor belongs to a parametric model. A new estimator for the unknown parameter is defined as the value that minimizes the distance between a vector of weighted integrals of the tail dependence function and their empirical counterparts. The minimization problem has, with probability tending to one, a unique, global solution. The estimator is consistent and asymptotically normal. The spectral measures of the tail dependence models to which the method applies can be discrete or continuous. Examples demonstrate the applicability and the performance of the method.

Article information

Source
Ann. Statist., Volume 40, Number 3 (2012), 1764-1793.

Dates
First available in Project Euclid: 2 October 2012

Permanent link to this document
https://projecteuclid.org/euclid.aos/1349196391

Digital Object Identifier
doi:10.1214/12-AOS1023

Mathematical Reviews number (MathSciNet)
MR3015043

Zentralblatt MATH identifier
1257.62058

Subjects
Primary: 62G32: Statistics of extreme values; tail inference 62G05: Estimation 62G10: Hypothesis testing 62G20: Asymptotic properties 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles 60G70: Extreme value theory; extremal processes

Keywords
Asymptotic statistics factor model M-estimation multivariate extremes tail dependence

Citation

Einmahl, John H. J.; Krajina, Andrea; Segers, Johan. An M-estimator for tail dependence in arbitrary dimensions. Ann. Statist. 40 (2012), no. 3, 1764--1793. doi:10.1214/12-AOS1023. https://projecteuclid.org/euclid.aos/1349196391


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References

  • Ballani, F. and Schlather, M. (2011). A construction principle for multivariate extreme value distributions. Biometrika 98 633–645.
  • Beirlant, J., Goegebeur, Y., Teugels, J. and Segers, J. (2004). Statistics of Extremes: Theory and Applications. Wiley, Chichester.
  • Boldi, M. O. and Davison, A. C. (2007). A mixture model for multivariate extremes. J. R. Stat. Soc. Ser. B Stat. Methodol. 69 217–229.
  • Coles, S. G. and Tawn, J. A. (1991). Modelling extreme multivariate events. J. R. Stat. Soc. Ser. B Stat. Methodol. 53 377–392.
  • Cooley, D., Davis, R. A. and Naveau, P. (2010). The pairwise beta distribution: A flexible parametric multivariate model for extremes. J. Multivariate Anal. 101 2103–2117.
  • de Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer, New York.
  • de Haan, L., Neves, C. and Peng, L. (2008). Parametric tail copula estimation and model testing. J. Multivariate Anal. 99 1260–1275.
  • de Haan, L. and Resnick, S. I. (1977). Limit theory for multivariate sample extremes. Z. Wahrsch. Verw. Gebiete 40 317–337.
  • Drees, H. and Huang, X. (1998). Best attainable rates of convergence for estimators of the stable tail dependence function. J. Multivariate Anal. 64 25–47.
  • Einmahl, J. H. J. (1997). Poisson and Gaussian approximation of weighted local empirical processes. Stochastic Process. Appl. 70 31–58.
  • Einmahl, J. H. J., Krajina, A. and Segers, J. (2008). A method of moments estimator of tail dependence. Bernoulli 14 1003–1026.
  • Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Applications of Mathematics (New York) 33. Springer, Berlin.
  • Fama, E. F. and French, K. R. (1993). Common risk factors in the returns on stocks and bonds. Journal of Financial Economics 33 3–56.
  • Fama, E. F. and French, K. R. (1996). Multifactor explanations of asset pricing anomalies. J. Finance 51 55–84.
  • Frees, E. W. and Valdez, E. A. (1998). Understanding relationships using copulas. N. Am. Actuar. J. 2 1–25.
  • Geluk, J. L., de Haan, L. and de Vries, C. G. (2007). Weak and strong financial fragility. Technical Report 2007-023/2, Tinbergen Institute.
  • Guillotte, S., Perron, F. and Segers, J. (2011). Non-parametric Bayesian inference on bivariate extremes. J. R. Stat. Soc. Ser. B Stat. Methodol. 73 377–406.
  • Gumbel, E. J. (1960). Bivariate exponential distributions. J. Amer. Statist. Assoc. 55 698–707.
  • Huang, X. (1992). Statistics of bivariate extreme values. Ph.D. thesis, Tinbergen Institute Research Series.
  • Joe, H., Smith, R. L. and Weissman, I. (1992). Bivariate threshold methods for extremes. J. R. Stat. Soc. Ser. B Stat. Methodol. 54 171–183.
  • Kleibergen, F. (2011). Reality checks for and of factor pricing. Technical report, Dept. Economics, Brown Univ. Preprint. Available at http://www.econ.brown.edu/fac/Frank_Kleibergen/.
  • Ledford, A. W. and Tawn, J. A. (1996). Statistics for near independence in multivariate extreme values. Biometrika 83 169–187.
  • Ledford, A. W. and Tawn, J. A. (1998). Concomitant tail behaviour for extremes. Adv. in Appl. Probab. 30 197–215.
  • Malevergne, Y. and Sornette, D. (2004). Tail dependence of factor models. Journal of Risk 6 71–116.
  • Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York.
  • Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Applied Probability. A Series of the Applied Probability Trust 4. Springer, New York.
  • Ribatet, M. (2011). POT: Generalized Pareto distribution and peaks over threshold. R package Version 1.1-1.
  • Smith, R. L. (1994). Multivariate threshold methods. In Extreme Value Theory and Applications (J. Galambos, J. Lechner and E. Simiu, eds.) 225–248. Kluwer Academic, Dordrecht.
  • Tawn, J. A. (1988). Bivariate extreme value theory: Models and estimation. Biometrika 75 397–415.
  • Vervaat, W. (1972). Functional central limit theorems for processes with positive drift and their inverses. Z. Wahrsch. Verw. Gebiete 23 245–253.
  • Wang, Y. and Stoev, S. A. (2011). Conditional sampling for max-stable random fields. Adv. in Appl. Probab. 43 463–481.