Abstract

In multivariate heavy tail estimation, the support of the limit measure

provides information on the asymptotic dependence structure of the random vector with the heavy tail distribution. This asymptotic dependence structure may

be difficult to discern, even in favorable cases of R2

+-valued data since exploratory

methods can be ambiguous and heavily dependent on threshold choice. We restrict ourselves to techniques that help distinguish between the following asymp-

totic models for heavy tails on R2

+: (i) full dependence where the limit measure

concentrates on a ray from the origin; (ii) strong dependence where the support

of the limit measure is a proper connected subcone of the positive quadrant; (iii)

weak dependence where the limit measure concentrates on the whole positive

quadrant. We propose two test statistics, analyze their asymptotically normal

behavior under full and not-full dependence, and discuss method implementation

using bootstrap methods. The methodology is illustrated with both simulated

and real data.

Information

Preprint No.SS-2024-0196
Manuscript IDSS-2024-0196
Complete AuthorsTiandong Wang, Sidney I. Resnick
Corresponding AuthorsTiandong Wang
Emailstd_wang@fudan.edu.cn

References

  1. Basrak, B. and H. Planini´c (2019). A note on vague convergence of measures. Statist. Probab. Lett. 153, 180–186.
  2. Bhattacharya, A., B. Chen, and R. van der Hofstad, B. Zwart (2020). Consistency of the PLFit estimator for power-law data. ArXiv eprint: 2002.06870.
  3. Bollob´as, B., C. Borgs, J. Chayes, , and O. Riordan (2003). Directed scale-free graphs. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms (Baltimore, MD, 2003), pp. 132–139. ACM, New York.
  4. Cirkovic, D., T. Wang, and S. Resnick (2023). Preferential attachment with reciprocity: Properties and estimation. Journal of Complex Networks 11(5), cnad031.
  5. Clauset, A., C. Shalizi, and M. Newman (2009). Power-law distributions in empirical data. SIAM Rev. 51(4), 661–703.
  6. Csardi, G. and T. Nepusz (2006). The igraph software package for complex network research.
  7. InterJournal, Complex Systems 1695(5), 1–9.
  8. Das, B. and M. Kratz (2020). Risk concentration under second order regular variation. Extremes 23(3), 381–410.
  9. Das, B., A. Mitra, and S. Resnick (2013). Living on the multi-dimensional edge: Seeking hidden risks using regular variation. Advances in Applied Probability 45(1), 139–163.
  10. Das, B. and S. Resnick (2017). Hidden regular variation under full and strong asymptotic dependence. Extremes 20(4), 873–904.
  11. Davis, R., L. Fernandes, and K. Fokianos (2023). Clustering multivariate time series using energy distance. Journal of Time Series Analysis 44(5-6), 487–504.
  12. de Haan, L. (1996). von Mises-type conditions in second order regular variation. J. Math. Anal. Appl. 197(2), 400–410.
  13. de Haan, L. and J. de Ronde (1998). Sea and wind: multivariate extremes at work. Extremes 1(1), 7–46.
  14. de Haan, L. and A. Ferreira (2006). Extreme Value Theory: An Introduction. New York: Springer-Verlag.
  15. de Haan, L. and S. Resnick (1993). Estimating the limit distribution of multivariate extremes. Stochastic Models 9(2), 275–309.
  16. de Haan, L. and S. Resnick (1996). Second-order regular variation and rates of convergence in extreme-value theory. Ann. Probab. 24(1), 97–124.
  17. de Haan, L. and U. Stadtmueller (1996). Generalized regular variation of second order. J. Aust. Math. Soc., Ser. A 61(3), 381–395.
  18. Drees, H., A. Janßen, and S. Neblung (2021). Cluster based inference for extremes of time series. Stochastic Process. Appl. 142, 1–33.
  19. Drees, H., A. Janßen, and Resnick, S.I., Wang, T. (2020). On a minimum distance procedure for threshold selection in tail analysis. Siam J. Math. Data Sci., 75–102.
  20. Drees, H. and A. Sabourin (2021). Principal component analysis for multivariate extremes. Electron. J. Stat. 15(1), 908–943.
  21. Einmahl, J., F. Yang, and C. Zhou (2021). Testing the multivariate regular variation model. J. Bus. Econom. Statist. 39(4), 907–919.
  22. Feigin, P. and S. Resnick (1997). Linear programming estimators and bootstrapping for heavy tailed phenomena. Adv. in Appl. Probab. 29, 759–805.
  23. Fomichov, V. and J. Ivanovs (2023). Spherical clustering in detection of groups of concomitant extremes. Biometrika 110(1), 135–153.
  24. Gillespie, C. (2015). Fitting heavy tailed distributions: The poweRlaw package. Journal of Statistical Software 64(2), 1–16. http://www.jstatsoft.org/v64/i02/.
  25. Hahn, M. (1978). Central limit theorems in D[0, 1]. Z. Wahrsch. Verw. Gebiete 44(2), 89–101.
  26. Hult, H. and F. Lindskog (2006). Regular variation for measures on metric spaces. Publ. Inst. Math. (Beograd) (N.S.) 80(94), 121–140.
  27. Janßen, A. and P. Wan (2020). k-means clustering of extremes. Electron. J. Stat. 14(1), 1211–1233.
  28. Krapivsky, P. and S. Redner (2001). Organization of growing random networks. Physical Review E 63(6), 066123:1–14.
  29. Kulik, R. and P. Soulier (2020). Heavy-Tailed Time Series. Springer Series in Operations Research and Financial Engineering. New York, NY: Springer.
  30. Lehtomaa, J. and S. Resnick (2020). Asymptotic independence and support detection techniques for heavy-tailed multivariate data. Insurance: Mathematics and Economics 93, 262 – 277.
  31. Lindskog, F., S. Resnick, and J. Roy (2014). Regularly varying measures on metric spaces: Hidden regular variation and hidden jumps. Probab. Surv. 11, 270–314.
  32. Peng, L. (1998). Second Order Condition and Extreme Value Theory. Ph. D. thesis, Tinbergen
  33. Institute, Erasmus University, Rotterdam.
  34. Pollard, D. (1990). Empirical Processes: Theory and Applications. NSF-CBMS Regional Conference Series in Probability and Statistics, Institute of Mathematical Statistics.
  35. Resnick, S. (2002). Hidden regular variation, second order regular variation and asymptotic
  36. independence. Extremes 5(4), 303–336 (2003).
  37. Resnick, S. (2007). Heavy Tail Phenomena: Probabilistic and Statistical Modeling. Springer Series in Operations Research and Financial Engineering. New York: Springer-Verlag. ISBN: 0-387-24272-4.
  38. Resnick, S. (2024). The Art of Finding Hidden Risks; Hidden Regular Variation in the 21st Century. Switzerland: Springer. ISBN: 978-3-031-57598-3.
  39. Samorodnitsky, G., S. Resnick, and Towsley, D, Davis, R, Willis, A, Wan, P (2016, March).
  40. Nonstandard regular variation of in-degree and out-degree in the preferential attachment model. Journal of Applied Probability 53(1), 146–161.
  41. Virkar, Y. and A. Clauset (2014). Power-law distributions in binned empirical data. Ann. Appl. Stat. 8(1), 89–119.
  42. Wang, T. and S. Resnick (2022a). Asymptotic dependence of in- and out-degrees in a preferential attachment model with reciprocity. Extremes 25, 417–450.
  43. Wang, T. and S. Resnick (2022b). Measuring reciprocity in a directed preferential attachment network. Adv. in Appl. Probab. 54(3), 718–742.
  44. Wang, T. and S. Resnick (2024). Random networks with heterogeneous reciprocity. Extremes 27, 123–161. Fudan University, and Shanghai Academy of Artificial Intelligence for Science

Acknowledgments

T. Wang gratefully acknowledges National Natural Science Foundation of

China Grant 12301660 and Science and Technology Commission of Shanghai Municipality Grant 23JC1400700.

The first author also thanks Shanghai Institute for Mathematics and

Interdisciplinary Sciences (SIMIS) for their financial support. This research

was partly funded by SIMIS under grant number SIMIS-ID-2024-WE. T.

Wang is grateful for the resources and facilities provided by SIMIS, which

were essential for the completion of this work.

Supplementary Materials

The online supplementary material contains additional simulation results

and technical details for all theoretical results in the main paper.

Supplementary materials are available for download.