Skip to main content
SpringerLink
Log in

Multivariate Generalized Pareto Distributions

  • Chapter
  • First Online:
Laws of Small Numbers: Extremes and Rare Events

  • 2335 Accesses

Abstract

In analogy to the univariate case, we introduce certain multivariate generalized Pareto df (GPD) of the form W = 1 + log(G) for the statistical modelling of multivariate exceedances, see Section 5.1. Various results around the multivariate peaks-over-threshold approach are compiled in Section 5.2. The peaks-overthreshold stability of a multivariate GPD is investigated in Section 5.3.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. Abdous, B., Ghoudi, K., and Khoudraji, A. (1999). Non-parametric estimation of the limit dependence function of multivariate extremes. Extremes 2, 245-268.

    Article  MATH  MathSciNet  Google Scholar 

  2. Alsina, C., Nelsen, R.B., and Schweizer, B. (1993). On the characterization of a class of binary operations on distribution functions. Statist. Probab. Letters 17, 85-89.

    Article  MATH  MathSciNet  Google Scholar 

  3. Aulbach, S., Bayer, V., and Falk, M. (2009). A multivariate piecing-together approach with an application to operational loss data. Technical report, University of Würzburg.

    Google Scholar 

  4. Bhattacharya, R.N., and Rao, R.R. (1976). Normal Approximation and Asymptotic Expansion. Wiley, New York.

    Google Scholar 

  5. Buishand, T.A., Haan, L. de, and Zhou, C. (2006). On spatial extremes: with application to a rainfall problem. Ann. Appl. Stat. 2, 624-642.

    Article  Google Scholar 

  6. Capéraà, P., Fougères, A.-L., and Genest, C. (1997). A nonparametric estimation procedure for bivariate extreme value copulas. Biometrika 84, 567- 577.

    Article  MATH  MathSciNet  Google Scholar 

  7. Coles, S.G. (2001). An Introduction to Statistical Modeling of Extreme Values. Springer Series in Statistics, Springer, New York.

    Google Scholar 

  8. Cuculescu, I., and Theodorescu, R. (2001). Copulas: diagonals, tracks. Rev. Roumaine Math. Pures Appl. 46, 731-742.

    MATH  MathSciNet  Google Scholar 

  9. Davison, A.C., and Smith, R.L. (1990). Models for exceedances over high thresholds (with discussions). J.. Roy. Statist. Soc., Ser. B 52, 393-442.

    MATH  MathSciNet  Google Scholar 

  10. Deheuvels, P. (1978). Caractèrisation complète des lois extrème multivarièes et de la convergene des types extrèmes. Publ. Inst. Statist. Univ. Paris 23, 1-36.

    MATH  MathSciNet  Google Scholar 

  11. Deheuvels, P. (1984). Probabilistic aspects of multiariate extremes. In Statistical Extremes and Applications (J. Tiago de Oliveira ed.), 117-130, D. Reidel Publishing Company.

    Google Scholar 

  12. Deheuvels, P. (1991). On the limiting behavior of the Pickands estimator for bivariate extreme-value distributions. Statist. Probab. Letters 12, 429-439.

    Article  MATH  MathSciNet  Google Scholar 

  13. Deheuvels, P., and Tiago de Oliveira, J. (1989). On the non-parametric estimation of the bivariate extreme-value distributions. Statist. Probab. Letters 8, 315-323.

    Article  MATH  MathSciNet  Google Scholar 

  14. Einmahl, J.H.J., Haan, L. de, and Piterbarg, V. (2001). Nonparametric Estimation of the Spectral Measure of an Extreme Value Distribution. Ann. Stat 29, 1401-1423.

    Article  MATH  Google Scholar 

  15. Falk, M. (1990). A note on generalized Pareto distributions and the k upper extremes. Probab. Th. Rel. Fields 85, 499-503.

    Article  MATH  MathSciNet  Google Scholar 

  16. Falk, M. (1998). Local asymptotic normality of truncated empirical processes. Ann. Statist. 26, 692-718.

    Article  MATH  MathSciNet  Google Scholar 

  17. Falk, M. (2003). Domain of multivariate attraction via angular decomposition. Comm. Statist. - Theory Methods 32, 2109-2116.

    Article  MATH  MathSciNet  Google Scholar 

  18. Falk, M., and Guillou, A. (2008). Peaks-over-threshold stability of multivariate generalized Pareto distributions. J. Multivar. Anal. 99, 715-734.

    Article  MATH  MathSciNet  Google Scholar 

  19. Falk, M., Marohn, F., and Tewes, B. (2002). Foundations of Statistical Analyses and Applications with SAS, Birkhäuser, Basel.

    MATH  Google Scholar 

  20. Falk, M., and Michel, R. (2009). Testing for a Multivariate Generalized Pareto Distribution. Extremes 12, 33-51.

    Article  MathSciNet  Google Scholar 

  21. Falk, M., and Reiss, R.-D. (2002). A characterization of the rate of convergence in bivariate extreme value models. Statist. Probab. Letters 59, 341-351.

    Article  MATH  MathSciNet  Google Scholar 

  22. Falk, M., and Reiss, R.-D. (2005). On Pickands coordinates in arbitrary dimensions. J. Mult. Analysis 92, 426-453.

    Article  MATH  MathSciNet  Google Scholar 

  23. Frick, M., and Reiss, R.-D. (2009). Expansions of multivariate Pickands densities and testing the tail dependence. J. Multivariate Anal. 100, 1168-1181.

    Article  MATH  MathSciNet  Google Scholar 

  24. Galambos, J. (1987). The Asymptotic Theory of Extreme Order Statistics. 2nd edition, Krieger, Malabar.

    Google Scholar 

  25. Genest, C., Quesada-Molina, J.J., Rodríguez-Lallena, J.A., and Sempi, C. (1999). A characterization of quasi-copulas. J. Mult. Analysis 69, 193-205.

    Article  MATH  Google Scholar 

  26. Ghoudi, K., Khoudraji, A., and Rivest, L.P. (1998). Statistical properties of couples of bivariate extreme-value copulas. Canad. J. Statist. 26, 187-197.

    Article  MATH  MathSciNet  Google Scholar 

  27. Haan, L. de (1985). Extremes in higher dimensions: The model and some statistics. In Proc. 45th Session ISI (Amsterdam), 26.3.

    Google Scholar 

  28. Haan, L. de, and Peng, L. (1997). Rates of convergence for bivariate extremes. J. Mult. Analysis 61, 195-230.

    Article  MATH  Google Scholar 

  29. Haan, L. de, and Ronde, J. de (1998). Sea and wind: multivariate extremes at work. Extremes 1, 7-45.

    Article  MATH  MathSciNet  Google Scholar 

  30. Hall, P. (1982 b). On some simple estimates of an exponent of regular variation. J. Royal Statist. Soc. B 44, 37-42.

    MATH  Google Scholar 

  31. Hall, P., and Tajvidi, N. (2000). Distribution and dependence function estimation for bivariate extreme-value distributions. Bernoulli 6, 835-844.

    Article  MATH  MathSciNet  Google Scholar 

  32. Hofmann, D. (2009). Characterization of the D-Norm Corresponding to a Multivariate Extreme Value Distribution. PhD-thesis, University of Würzburg.

    Google Scholar 

  33. Huang, X. (1992). Statistics of Bivariate Extreme Values. PhD-thesis, Tinbergen Institute Research Series.

    Google Scholar 

  34. Jiménez, J.R., Villa-Diharce, E., and Flores, M. (2001). Nonparametric estimation of the dependence function in bivariate extreme value distributions. J. Mult. Analysis 76, 159-191.

    Article  MATH  Google Scholar 

  35. Juri, A., and Wüthrich, M.V. (2004). Tail dependence from a distributional point of view. Extremes 6, 213-246.

    Article  Google Scholar 

  36. Kaufmann, E., and Reiss, R.-D. (1995). Approximation rates for multivariate exceedances. J. Statist. Plann. Inference 45, 235-245.

    Article  MATH  MathSciNet  Google Scholar 

  37. Kotz, S., and Nadarajah, S. (2000). Extreme Value Distributions. Theory and Applications. Imperial College Press, London.

    Book  MATH  Google Scholar 

  38. Mardia, K.V., Kent, J., and Bibby, J. (1979). Multivariate Analysis. Academic, London.

    MATH  Google Scholar 

  39. Michel, R. (2006). Simulation and Estimation in Multivariate Generalized Pareto Models. PhD-thesis, University of Würzburg. http://www.opusbayern.de/uni-wuerzburg/volltexte/2006/1848/

  40. Michel, R. (2007). Simulation of certain multivariate generalized Pareto distributions. Extremes 10, 83-107.

    Article  MATH  MathSciNet  Google Scholar 

  41. Michel, R. (2009). Parametric estimation procedures in multivariate generalized Pareto models. Scan. Jour. Stat. 36, 60-75.

    MATH  Google Scholar 

  42. Nadarajah, S. (2000). Approximation for bivariate extreme values. Extremes 3, 87-98.

    Article  MATH  MathSciNet  Google Scholar 

  43. Nelsen, R.B. (2006). An Introduction to Copulas. 2nd. ed., Springer, New York.

    MATH  Google Scholar 

  44. Omey, E., and Rachev, S.T. (1991). Rates of convergence in multivariate extreme value theory. J. Mult. Analysis 38, 36-50.

    Article  MATH  MathSciNet  Google Scholar 

  45. Pfanzagl, J. (1994). Parametric Statistical Theory. De Gruyter, Berlin.

    MATH  Google Scholar 

  46. Pickands III, J. (1981). Multivariate extreme value distributions. Proc. 43th Session ISI (Buenos Aires), pp. 859-878.

    Google Scholar 

  47. Reiss, R.-D. (1989). Approximate Distributions of Order Statistics. (With Applications to Nonparametric Statistics). Springer Series in Statistics, Springer, New York.

    Google Scholar 

  48. Reiss, R.-D., and Thomas, M. (2007). Statistical Analysis of Extreme Values, 3rd ed., Birkhäuser, Basel.

    MATH  Google Scholar 

  49. Resnick, S.I. (1987). Extreme Values, Regular Variation, and Point Processes. Applied Prob. Vol. 4, Springer, New York.

    Google Scholar 

  50. Rootzén, H., and Tajvidi, N. (2006). Multivariate generalized Pareto distributions. Bernoulli 12, 917-930.

    Article  MATH  MathSciNet  Google Scholar 

  51. Serfling, R.J. (1980). Approximation Theorems of Mathematical Statistics. Wiley, New York.

    Book  MATH  Google Scholar 

  52. Sheskin, D.J. (2003). Parametric and Nonparametric Statistical Procedures. 3rd ed., Chapman&Hall/CRC, Boca Raton.

    Google Scholar 

  53. Shi, D. (1995). Multivariate extreme value distribution and its Fisher information matrix. ACTA Math. Appl. Sinica 11, 421-428.

    Article  MATH  Google Scholar 

  54. Smith, R.L., Tawn, J.A., and Yuen, H.K. (1990). Statistics of multivariate extremes. Int. Statist. Review 58, 47-58.

    Article  MATH  Google Scholar 

  55. Stephenson, A. (2003). Simulating multivariate extreme value distributions of logistic type. Extremes 6, 49-59.

    Article  MATH  MathSciNet  Google Scholar 

  56. Tawn, J. (1990). Modelling multivariate extreme value distributions. Biometrika 77, 245-253.

    Article  MATH  Google Scholar 

  57. Tiago de Oliveira, J. (1989 a). Intrinsic estimation of the dependence structure for bivariate extremes. Statist. Probab. Letters 8, 213-218.

    Article  MATH  MathSciNet  Google Scholar 

  58. Wüthrich, M.V. (2004). Bivariate extension of the Pickands-Balkema-de Haan theorem. Annales de L’Institut Henry Poincaré-Probabilités & Statistiques 40, 33-41.

    MATH  Google Scholar 

  59. Zhang, D., Wells, M.T., and Peng, L. (2008). Nonparametric estimation of the dependence function for a multivariate extreme value distribution. J. Mult. Analysis 99, 577-588.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, University of Würzburg, Am Hubland, 97074, Würzburg, Germany

    Michael Falk

  2. Department of Mathematical Statistics and Actuarial Science, University of Berne, Sidlerstrasse 5, 3012, Bern, Switzerland

    Jürg Hüsler

  3. Department of Mathematics, University of Siegen, Walter Flex Str. 3, 57068, Siegen, Germany

    Rolf-Dieter Reiss

Authors
  1. Michael Falk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jürg Hüsler
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rolf-Dieter Reiss
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael Falk .

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Basel AG

About this chapter

Cite this chapter

Falk, M., Hüsler, J., Reiss, RD. (2011). Multivariate Generalized Pareto Distributions. In: Laws of Small Numbers: Extremes and Rare Events. Springer, Basel. https://doi.org/10.1007/978-3-0348-0009-9_5

Download citation

Publish with us

Policies and ethics

Search

Navigation