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Extremes

, Volume 20, Issue 4, pp 873–904 | Cite as

Hidden regular variation under full and strong asymptotic dependence

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Abstract

Data exhibiting heavy-tails in one or more dimensions is often studied using the framework of regular variation. In a multivariate setting this requires identifying specific forms of dependence in the data; this means identifying that the data tends to concentrate along particular directions and does not cover the full space. This is observed in various data sets from finance, insurance, network traffic, social networks, etc. In this paper we discuss the notions of full and strong asymptotic dependence for bivariate data along with the idea of hidden regular variation in these cases. In a risk analysis setting, this leads to improved risk estimation accuracy when regular methods provide a zero estimate of risk. Analyses of both real and simulated data sets illustrate concepts of generation and detection of such models.

Keywords

Regular variation Multivariate heavy tails Hidden regular variation Tail estimation Strong dependence 

AMS 2000 Subject Classifications

28A33 60G70 62G05 62G32 

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References

  1. Anderson, P.L., Meerschaert, M.M.: Modeling river flows with heavy tails. Water Resour. Res. 34(9), 2271–2280 (1998)CrossRefGoogle Scholar
  2. Beirlant, J., Vynckier, P., Teugels, J.: Tail index estimation, Pareto quantile plots, and regression diagnostics. J. Amer. Statist. Assoc. 91(436), 1659–1667 (1996)MathSciNetMATHGoogle Scholar
  3. Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular variation, Volume 27 of Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (1989)MATHGoogle Scholar
  4. Bollobás, B., Borgs, C., Chayes, J., Riordan, O.: Directed Scale-Free Graphs Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms (Baltimore, 2003), pp. 132–139. ACM, New York (2003)Google Scholar
  5. Crovella, M., Bestavros, A., Taqqu, M.S.: Heavy-tailed probability distributions in the world wide web. In: Taqqu, M.S., Adler, R., Feldman, R. (eds.) A Practical Guide to Heavy Tails: Statistical Techniques for Analysing Heavy Tailed Distributions. Birkhäuser, Boston (1999)Google Scholar
  6. Csardi, G., Nepusz, T.: The igraph software package for complex network research. Interjournal. Comput. Syst. 1695(5), 1–9 (2006)Google Scholar
  7. Das, B., Embrechts, P., Fasen, V.: Four theorems and a financial crisis. Int. J. Approx. Reason. 54(6), 701–716 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. Das, B., Mitra, A., Resnick, S.I.: Living on the multidimensional edge: seeking hidden risks using regular variation. Adv. Appl. Probab. 45(1), 139–163 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. Das, B., Resnick, S.I.: Conditioning on an extreme component Model consistency with regular variation on cones. Bernoulli 17(1), 226–252 (2011)MathSciNetCrossRefMATHGoogle Scholar
  10. Das, B., Resnick, S.I.: Detecting a conditional extreme value model. Extremes 14(1), 29–61 (2011)MathSciNetCrossRefMATHGoogle Scholar
  11. Das, B., Resnick, S.I.: Models with hidden regular variation: generation and detection. Stochastic Systems 5, 195–238 (2015)Google Scholar
  12. de Haan, L., Ferreira, A.: Extreme Value Theory: an Introduction. Springer-Verlag, New York (2006)Google Scholar
  13. Drees, H., de Haan, L., Resnick, S.I.: How to make a Hill plot. Ann. Statist. 28(1), 254–274 (2000)Google Scholar
  14. Durrett, R.T.: Random Graph Dynamics. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (2010)Google Scholar
  15. Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extreme Events for Insurance and Finance. Springer-Verlag, Berlin (1997)CrossRefMATHGoogle Scholar
  16. Heffernan, J.E., Resnick, S.I.: Hidden regular variation and the rank transform. Adv. Appl. Probab. 37(2), 393–414 (2005)MathSciNetCrossRefMATHGoogle Scholar
  17. Heffernan, J.E., Resnick, S.I.: Limit laws for random vectors with an extreme component. Ann. Appl. Probab. 17(2), 537–571 (2007)MathSciNetCrossRefMATHGoogle Scholar
  18. Heffernan, J.E., Tawn, J.A.: A conditional approach for multivariate extreme values (with discussion). J. R. Stat. Soc. Ser. B 66(3), 497–546 (2004)MathSciNetCrossRefMATHGoogle Scholar
  19. Hult, H., Lindskog, F.: Regular variation for measures on metric spaces. Publications de l’Institut mathématique Nouvelle Série 80(94), 121–140 (2006)MathSciNetMATHGoogle Scholar
  20. Ibragimov, R., Jaffee, D., Walden, J.: Diversification disasters. J. Financ. Econ. 99(2), 333–348 (2011)CrossRefGoogle Scholar
  21. Kratz, M., Resnick, S.I.: The qq–estimator and heavy tails. Stoch. Model. 12, 699–724 (1996)MathSciNetCrossRefMATHGoogle Scholar
  22. Lindskog, F., Resnick, S.I., Roy, J.: Regularly varying measures on metric spaces: hidden regular variation and hidden jumps. Probab. Surv. 11, 270–314 (2014)MathSciNetCrossRefMATHGoogle Scholar
  23. Resnick, S.I.: Hidden regular variation, second order regular variation and asymptotic independence. Extremes 5(4), 303–336 (2002)MathSciNetCrossRefMATHGoogle Scholar
  24. Resnick, S.I.: Heavy Tail Phenomena: Probabilistic and Statistical Modeling. Springer Series in Operations Research and Financial Engineering. Springer-Verlag, New York (2007)Google Scholar
  25. Resnick, S.I.: Extreme Values, Regular Variation and Point Processes. Springer Series in Operations Research and Financial Engineering. Springer, New York (2008). Reprint of the 1987 originalGoogle Scholar
  26. Resnick, S.I.: Multivariate regular variation on cones: application to extreme values, hidden regular variation and conditioned limit laws. Stochastics: An International Journal of Probability and Stochastic Processes 80, 269–298 (2008)MathSciNetCrossRefMATHGoogle Scholar
  27. Resnick, S.I., Samorodnitsky, G.: Tauberian theory for multivariate regularly varying distributions with application to preferential attachment networks. Extremes 18(3), 349–367 (2015)MathSciNetCrossRefMATHGoogle Scholar
  28. Resnick, S.I., Stărică, C.: Smoothing the Hill estimator. Adv. Appl. Probab. 29, 271–293 (1997)MathSciNetCrossRefMATHGoogle Scholar
  29. Samorodnitsky, G., Resnick, S., Towsley, D., Davis, R.,Willis, A.,Wan, P.: Nonstandard regular variation of in-degree and out-degree in the preferential attachment model. J. Appl. Probab. 53(1), 146–161 (2016)Google Scholar
  30. Smith, R.L.: Statistics of extremes, with applications in environment, insurance and finance. In: Finkenstadt, B., Rootzén, H. (eds.) SemStat: Seminaire Europeen de Statistique, Exteme Values in Finance, Telecommunications, and the Environment , pp. 1–78. Chapman-Hall, London (2003)Google Scholar
  31. Viswanath, B., Mislove, A., Cha, M., Gummadi, K.P.: On the evolution of user interaction in facebook Proceedings of the 2nd ACM SIGCOMM Workshop on Social Networks (WOSN’09) (2009)Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Engineering Systems and DesignSingapore University of Technology and DesignSingaporeSingapore
  2. 2.School of ORIECornell UniversityIthacaUSA

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