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Extremes

pp 1–30 | Cite as

Identifying groups of variables with the potential of being large simultaneously

  • Maël Chiapino
  • Anne Sabourin
  • Johan Segers
Article
  • 9 Downloads

Abstract

Identifying groups of variables that may be large simultaneously amounts to finding out which joint tail dependence coefficients of a multivariate distribution are positive. The asymptotic distribution of a vector of nonparametric, rank-based estimators of these coefficients justifies a stopping criterion in an algorithm that searches the collection of all possible groups of variables in a systematic way, from smaller groups to larger ones. The issue that the tolerance level in the stopping criterion should depend on the size of the groups is circumvented by the use of a conditional tail dependence coefficient. Alternatively, such stopping criteria can be based on limit distributions of rank-based estimators of the coefficient of tail dependence, quantifying the speed of decay of joint survival functions. A performance score calculated by ten-fold cross-validation allows the user to select one among the various algorithms and set its tuning parameters in a data-driven way.

Keywords

Multivariate extremes Asymptotic dependence Statistical tests High dimensional data 

AMS 2000 Subject Classifications

MSC 62G32 MSC 62H15 MSC 62H05 MSC 62H20 

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Notes

References

  1. Agrawal, R., Srikant, R., et al.: Fast algorithms for mining association rules. In: Proceedings of the 20th International Conference on Very Large Data Bases, VLDB, vol. 1215, pp. 487–499 (1994)Google Scholar
  2. Bacro, J.N., Toulemonde, G.: Measuring and modelling multivariate and spatial dependence of extremes. J. Soc. Française Stat. 154(2), 139–155 (2013)MathSciNetzbMATHGoogle Scholar
  3. Bücher, A., Dette, H.: Multiplier bootstrap of tail copulas with applications. Bernoulli 19(5A), 1655–1687 (2013)MathSciNetCrossRefGoogle Scholar
  4. Chiapino, M., Sabourin, A.: Feature clustering for extreme events analysis, with application to extreme stream-flow data. In: ECML-PKDD 2016, Workshop NFmcp2016 (2016)Google Scholar
  5. Coles, S.G.: Regional modelling of extreme storms via max-stable processes. J. R. Stat. Soc. Ser. B (Methodol.) 55(4), 797–816 (1993)MathSciNetzbMATHGoogle Scholar
  6. Coles, S., Heffernan, J., Tawn, J.: Dependence measures for extreme value analyses. Extremes 2(4), 339–365 (1999)CrossRefGoogle Scholar
  7. De Haan, L., Zhou, C.: Extreme residual dependence for random vectors and processes. Adv. Appl. Probab. 43(01), 217–242 (2011)MathSciNetCrossRefGoogle Scholar
  8. Draisma, G., Drees, H., Ferreira, A., de Haan, L. Tail dependence in independence. Eurandom preprint (2001)Google Scholar
  9. Draisma, G., Drees, H., Ferreira, A., de Haan, L.: Bivariate tail estimation: dependence in asymptotic independence. Bernoulli 10(2), 251–280 (2004)Google Scholar
  10. Drees, H.: A general class of estimators of the extreme value index. J. Stat. Plan. Inference 66(1), 95–112 (1998a)MathSciNetCrossRefGoogle Scholar
  11. Drees, H.: On smooth statistical tail functionals. Scand. J. Stat. 25(1), 187–210 (1998b)MathSciNetCrossRefGoogle Scholar
  12. Eastoe, E.F., Tawn, J.A.: Modelling the distribution of the cluster maxima of exceedances of subasymptotic thresholds. Biometrika 99(1), 43–55 (2012)Google Scholar
  13. Einmahl, J.H.: Poisson and Gaussian approximation of weighted local empirical processes. Stoch. Process. Appl. 70(1), 31–58 (1997)MathSciNetCrossRefGoogle Scholar
  14. Einmahl, J.H., Krajina, A., Segers, J., et al.: An M-estimator for tail dependence in arbitrary dimensions. Ann. Stat. 40(3), 1764–1793 (2012)MathSciNetCrossRefGoogle Scholar
  15. Goix, N., Sabourin, A., Clémençon, S.: Sparse representation of multivariate extremes with applications to anomaly ranking. In: Proceedings of the 19th AISTAT Conference, pp 287–295 (2016)Google Scholar
  16. Goix, N., Sabourin, A., Clémençon, S.: Sparse representation of multivariate extremes with applications to anomaly detection. J. Multivar. Anal. 161, 12–31 (2017)MathSciNetCrossRefGoogle Scholar
  17. Ledford, A.W., Tawn, J.A.: Statistics for near independence in multivariate extreme values. Biometrika 83(1), 169–187 (1996)MathSciNetCrossRefGoogle Scholar
  18. Peng, L.: Estimation of the coefficient of tail dependence in bivariate extremes. Stat. Probab. Lett. 43(4), 399–409 (1999)MathSciNetCrossRefGoogle Scholar
  19. Pickands, III J.: Statistical inference using extreme order statistics. Ann. Stat. 3(1), 119–131 (1975)Google Scholar
  20. Qi, Y.: Almost sure convergence of the stable tail empirical dependence function in multivariate extreme statistics. Acta Math. Appl. Sin. (English series) 13(2), 167–175 (1997)MathSciNetCrossRefGoogle Scholar
  21. Ramos, A., Ledford, A.: A new class of models for bivariate joint tails. J. R. Stat. Soc. Ser. B 71(1), 219–241 (2009)MathSciNetCrossRefGoogle Scholar
  22. Resnick, S.I.: Heavy-Tail phenomena. In: Springer Series in Operations Research and Financial Engineering. Springer, New York (2007)Google Scholar
  23. Resnick, S.I.: Extreme Values, Regular Variation and Point Processes. Springer Series in Operations Research and Financial Engineering. Springer, New York, reprint of the 1987 original (2008)Google Scholar
  24. Rockafellar, R.T.: Convex Analysis. Princeton Mathematical Series, No. 28. Princeton University Press, Princeton (1970)Google Scholar
  25. Schlather, M., Tawn, J.A.: Inequalities for the extremal coefficients of multivariate extreme value distributions. Extremes 5(1), 87–102 (2002)MathSciNetCrossRefGoogle Scholar
  26. Schlather, M., Tawn, J.A.: A dependence measure for multivariate and spatial extreme values: Properties and inference. Biometrika 90(1), 139–156 (2003)MathSciNetCrossRefGoogle Scholar
  27. Shorack, G.R., Wellner, J.A.: Empirical Processes with Applications to Statistics. SIAM, Philadelphia (2009)CrossRefGoogle Scholar
  28. Smith, R.: Max-stable processes and spatial extremes. Unpublished manuscript (1990)Google Scholar
  29. Stephenson, A.: Simulating multivariate extreme value distributions of logistic type. Extremes 6(1), 49–59 (2003)MathSciNetCrossRefGoogle Scholar
  30. Tawn, J.A.: Modelling multivariate extreme value distributions. Biometrika 77 (2), 245–253 (1990)MathSciNetCrossRefGoogle Scholar
  31. van der Vaart, A.W.: Asymptotic Statistics, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 3. Cambridge University Press, Cambridge (1998)Google Scholar
  32. van der Vaart, A.W., Wellner, J.A.: Weak Convergence and Empirical Processes. Springer, New York (1996)CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.LTCI, Télécom ParisTechUniversité Paris-SaclayParisFrance
  2. 2.Institut de Statistique, Biostatistique et Sciences ActuariellesUniversité Catholique de LouvainLouvain-la-NeuveBelgium

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