Advertisement

Extremes

, Volume 14, Issue 1, pp 29–61 | Cite as

Detecting a conditional extreme value model

  • Bikramjit DasEmail author
  • Sidney I. Resnick
Article

Abstract

In classical extreme value theory probabilities of extreme events are estimated assuming all the components of a random vector to be in a domain of attraction of an extreme value distribution. In contrast, the conditional extreme value model assumes a domain of attraction condition on a sub-collection of the components of a multivariate random vector. This model has been studied in Heffernan and Tawn (JRSS B 66(3):497–546, 2004), Heffernan and Resnick (Ann Appl Probab 17(2):537–571, 2007), and Das and Resnick (2009). In this paper we propose three statistics which act as tools to detect this model in a bivariate set-up. In addition, the proposed statistics also help to distinguish between two forms of the limit measure that is obtained in the model.

Keywords

Regular variation Domain of attraction Heavy tails Asymptotic independence Conditional extreme value model 

AMS 2000 Subject Classifications

Primary—62G32 Secondary—60G70 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968)zbMATHGoogle Scholar
  2. Coles, S.G., Tawn, J.A.: Modelling extreme multivariate events. J. R. Stat. Soc., B 53, 377–392 (1991)MathSciNetzbMATHGoogle Scholar
  3. Coles, S.G., Heffernan, J.E., Tawn, J.A.: Dependence measures for extreme value analyses. Extremes 2(4), 339–365 (1999)zbMATHCrossRefGoogle Scholar
  4. Das, B., Resnick, S.I.: QQ plots, random sets and data from a heavy tailed distribution. Stoch. Models 24(1), 103–132 (2008). ISSN 1532-6349MathSciNetzbMATHCrossRefGoogle Scholar
  5. Das, B., Resnick, S.I.: Conditioning on an extreme component: model consistency and regular variation on cones. http://arxiv.org/abs/0805.4373 (2009)
  6. de Haan, L., de Ronde, J.: Sea and wind: multivariate extremes at work. Extremes 1(1), 7–46 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  7. de Haan, L., Ferreira, A.: Extreme Value Theory: an Introduction. Springer, New York (2006)zbMATHGoogle Scholar
  8. de Haan, L., Resnick, S.I.: Limit theory for multivariate sample extremes. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 40, 317–337 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  9. Fougères, A., Soulier, P.: Estimation of conditional laws given an extreme component. http://arXiv.org/0806.2426v2 (2009)
  10. Heffernan, J.E., Resnick, S.I.: Limit laws for random vectors with an extreme component. Ann. Appl. Probab. 17(2), 537–571 (2007). ISSN 1050–5164. doi: 10.1214/105051606000000835 MathSciNetzbMATHCrossRefGoogle Scholar
  11. Heffernan, J.E., Tawn, J.A.: A conditional approach for multivariate extreme values (with discussion). JRSS B 66(3), 497–546 (2004)MathSciNetzbMATHGoogle Scholar
  12. Hill, B.M.: A simple general approach to inference about the tail of a distribution. Ann. Stat. 3, 1163–1174 (1975)zbMATHCrossRefGoogle Scholar
  13. Kallenberg, O.: Random Measures, 3rd edn. Akademie-Verlag, Berlin (1983). ISBN 0-12-394960-2zbMATHGoogle Scholar
  14. Ledford, A.W., Tawn, J.A.: Statistics for near independence in multivariate extreme values. Biometrika 83(1), 169–187 (1996). ISSN 0006-3444MathSciNetzbMATHCrossRefGoogle Scholar
  15. Ledford, A.W., Tawn, J.A.: Modelling dependence within joint tail regions. J. R. Stat. Soc., Ser. B 59(2), 475–499 (1997). ISSN 0035-9246MathSciNetzbMATHCrossRefGoogle Scholar
  16. Ledford, A.W., Tawn, J.A.: Concomitant tail behaviour for extremes. Adv. Appl. Probab. 30(1), 197–215 (1998). ISSN 0001-8678MathSciNetzbMATHCrossRefGoogle Scholar
  17. López-Oliveros, L., Resnick, S.I.: Extremal dependence analysis of network sessions. Extremes (2009). http://arxiv.org/pdf/0905.1983v1
  18. Mason, D.: Laws of large numbers for sums of extreme values. Ann. Probab. 10, 754–764 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  19. Maulik, K., Resnick, S.I.: Characterizations and examples of hidden regular variation. Extremes 7(1), 31–67 (2005)MathSciNetCrossRefGoogle Scholar
  20. Maulik, K., Resnick, S.I., Rootzén, H.: Asymptotic independence and a network traffic model. J. Appl. Probab. 39(4), 671–699 (2002). ISSN 0021-9002MathSciNetzbMATHCrossRefGoogle Scholar
  21. McNeil, A.J., Frey, R., Embrechts, P.: Concepts, techniques and tools. In: Quantitative Risk Management. Princeton Series in Finance. Princeton University Press, Princeton (2005). ISBN 0-691-12255-5Google Scholar
  22. Neveu, J.: Processus ponctuels. In: École d’Été de Probabilités de Saint-Flour, VI—1976. Lecture Notes in Math., vol. 598, pp. 249–445. Springer, Berlin (1977)Google Scholar
  23. Pickands, J.: Statistical inference using extreme order statistics. Ann. Stat. 3, 119–131 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  24. Resnick, S.I.: A Probability Path. Birkhäuser, Boston (1999)zbMATHGoogle Scholar
  25. Resnick, S.I.: Hidden regular variation, second order regular variation and asymptotic independence. Extremes 5(4), 303–336 (2002)MathSciNetCrossRefGoogle Scholar
  26. Resnick. S.I.: Modeling data networks. In: Finkenstadt, B., Rootzén, H. (eds.) SemStat: Seminaire Europeen de Statistique, Extreme Values in Finance, Telecommunications, and the Environment, pp. 287–372. Chapman-Hall, London (2003)Google Scholar
  27. Resnick, S.I.: Heavy Tail Phenomena: Probabilistic and Statistical Modeling. Springer Series in Operations Research and Financial Engineering. Springer, New York (2007). ISBN: 0-387-24272-4zbMATHGoogle Scholar
  28. Resnick, S.I.: Extreme Values, Regular Variation and Point Processes. Springer Series in Operations Research and Financial Engineering. Springer, New York (2008). ISBN 978-0-387-75952-4. Reprint of the 1987 originalzbMATHGoogle Scholar
  29. Resnick, S.I., Stărică, C.: Consistency of Hill’s estimator for dependent data. J. Appl. Probab. 32(1), 139–167 (1995). ISSN 0021-9002MathSciNetzbMATHCrossRefGoogle Scholar
  30. Sarvotham, S., Riedi, R., Baraniuk, R.: Network and user driven on–off source model for network traffic. Comput. Networks 48, 335–350 (2005) (Special issue on “Long-range Dependent Traffic”)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.School of Operations Research and Information EngineeringCornell UniversityIthacaUSA

Personalised recommendations