Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-23T16:15:45.606Z Has data issue: false hasContentIssue false
  • English
  • Français

Sparse regular variation

Part of: Multivariate analysis Stochastic processes

Published online by Cambridge University Press:  22 November 2021

Nicolas Meyer Open the ORCID record for Nicolas Meyer [Opens in a new window]  and
Olivier Wintenberger
Nicolas Meyer*
Affiliation:
Sorbonne Université
Olivier Wintenberger*
Affiliation:
Sorbonne Université
*
*Postal address: Sorbonne Université, LPSM, 4 place Jussieu, F-75005, Paris, France.
*Postal address: Sorbonne Université, LPSM, 4 place Jussieu, F-75005, Paris, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

Regular variation provides a convenient theoretical framework for studying large events. In the multivariate setting, the spectral measure characterizes the dependence structure of the extremes. This measure gathers information on the localization of extreme events and often has sparse support since severe events do not simultaneously occur in all directions. However, it is defined through weak convergence, which does not provide a natural way to capture this sparsity structure. In this paper, we introduce the notion of sparse regular variation, which makes it possible to better learn the dependence structure of extreme events. This concept is based on the Euclidean projection onto the simplex, for which efficient algorithms are known. We prove that under mild assumptions sparse regular variation and regular variation are equivalent notions, and we establish several results for sparsely regularly varying random vectors.

Keywords

Multivariate extremesregular variationspectral measure

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 62H05: Characterization and structure theory
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 4 , December 2021 , pp. 1115 - 1148
Check for updates
Copyright
© The Author(s) 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Beirlant, J., Goegebeur, Y., Segers, J. and Teugels, J. L. (2006). Statistics of Extremes: Theory and Applications. John Wiley, Chichester.Google Scholar
Billingsley, P. (1995). Probability and Measure, 3rd edn. John Wiley, Hoboken, NJ.Google Scholar
Boldi, M.-O. and Davison, A. C. (2007). A mixture model for multivariate extremes. J. R. Statist. Soc. B [Statist. Methodology] 69, 217–229.CrossRefGoogle Scholar
Chautru, E. (2015). Dimension reduction in multivariate extreme value analysis. Electron. J. Statist. 9, 383–418.CrossRefGoogle Scholar
Chiapino, M. and Sabourin, A. (2016). Feature clustering for extreme events analysis, with application to extreme stream-flow data. In New Frontiers in Mining Complex Patterns (NFMCP 2016), Springer, Cham, pp. 132–147.Google Scholar
Chiapino, M., Sabourin, A. and Segers, J. (2019). Identifying groups of variables with the potential of being large simultaneously. Extremes 22, 193–222.CrossRefGoogle Scholar
Coles, S. G. and Tawn, J. A. (1991). Modelling extreme multivariate events. J. R. Statist. Soc. B [Statist. Methodology] 53, 377–392.Google Scholar
Condat, L. (2016). Fast projection onto the simplex and the $\ell_1$ ball. Math. Program. 158, 575–585.CrossRefGoogle Scholar
Cooley, D. and Thibaud, E. (2019). Decompositions of dependence for high-dimensional extremes. Biometrika 106, 587–604.CrossRefGoogle Scholar
Duchi, J., Shalev-Shwartz, S., Singer, Y. and Chandra, T. (2008). Efficient projections onto the $\ell_1$ -ball for learning in high dimensions. In Proceedings of the 25th International Conference on Machine Learning (ICML ’08), Association for Computing Machinery, New York, pp. 272–279.CrossRefGoogle Scholar
Einmahl, J., de Haan, L. and Huang, X. (1993). Estimating a multidimensional extreme-value distribution. J. Multivariate Anal. 47, 35–47.CrossRefGoogle Scholar
Einmahl, J., de Haan, L. and Sinha, A.K. (1997). Estimating the spectral measure of an extreme value distribution. Stoch. Process. Appl. 70, 143–171.CrossRefGoogle Scholar
Einmahl, J., de Haan, L. and Piterbarg, V. (2001). Nonparametric estimation of the spectral measure of an extreme value distribution. Ann. Statist. 29, 1401–1423.CrossRefGoogle Scholar
Einmahl, J. and Segers, J. (2009). Maximum empirical likelihood estimation of the spectral measure of an extreme-value distribution. Ann. Statist. 37, 2953–2989.CrossRefGoogle Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (2013). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.Google Scholar
Engelke, S. and Ivanovs, J. (2021). Sparse structures for multivariate extremes. To appear in Annual Rev. Statist. Appl. 8.CrossRefGoogle Scholar
Goix, N., Sabourin, A. and ClémenÇon, S. (2016). Sparse representation of multivariate extremes with applications to anomaly ranking. In Proc. Mach. Learning Res. 51, 75–83.Google Scholar
Goix, N., Sabourin, A. and ClémenÇon, S. (2017). Sparse representation of multivariate extremes with applications to anomaly detection. J. Multivariate Anal. 161, 12–31.CrossRefGoogle Scholar
Guillotte, S., Perron, F. and Segers, J. (2011). Non-parametric Bayesian inference on bivariate extremes. J. R. Statist. Soc. B [Statist. Methodology] 73, 377–406.CrossRefGoogle Scholar
De Haan, L. and Ferreira, A. (2006). Extreme Value Theory: an Introduction. Springer, New York.CrossRefGoogle Scholar
Janssen, A. and Wan, P. (2020). k-means clustering of extremes. Electron. J. Statist. 14, 1211–1233.CrossRefGoogle Scholar
Kyrillidis, A., Becker, S., Cevher, V. and Koch, C. (2013). Sparse projections onto the simplex. Internat. Conf. Mach. Learning 28, 235–243.Google Scholar
Ledford, A. W. and Tawn, J. A (1996). Statistics for near independence in multivariate extreme values. Biometrika 83, 169–187.CrossRefGoogle Scholar
Lehtomaa, J. and Resnick, S. I. (2020). Asymptotic independence and support detection techniques for heavy-tailed multivariate data. Insurance Math. Econom. 93, 262–277.CrossRefGoogle Scholar
Liu, J. and Ye, J. (2009). Efficient Euclidean projections in linear time. In Proc. 26th Annual International Conference on Machine Learning, ICML 2009, Association for Computing Machinery, New York, pp. 657–664.Google Scholar
Ramos, A. and Ledford, A. W. (2009). A new class of models for bivariate joint tails. J. R. Statist. Soc. B [Statist. Methodology] 71, 219–241.CrossRefGoogle Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation and Point Processes. Springer, New York.CrossRefGoogle Scholar
Resnick, S. I. (2007). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York.Google Scholar
Sabourin, A. and Drees, H. (2019). Principal component analysis for multivariate extremes. Preprint. Available at https://arxiv.org/abs/1906.11043.Google Scholar
Sabourin, A. and Naveau, P. (2014). Bayesian Dirichlet mixture model for multivariate extremes: a re-parametrization. Comput. Statist. Data Anal. 71, 542–567.CrossRefGoogle Scholar
Sabourin, A., Naveau, P. and Fougeres, A.-L. (2013). Bayesian model averaging for multivariate extremes. Extremes 16, 325–350.CrossRefGoogle Scholar
Segers, J. (2012). Max-stable models for multivariate extremes. REVSTAT Statist. J. 10, 61–82.Google Scholar
Sibuya, M. (1960). Bivariate extreme statistics. Ann. Inst. Statistic. Math. 11, 195–210.CrossRefGoogle Scholar
Simpson, E., Wadsworth, J. L and Tawn, J. A. (2020). Determining the dependence structure of multivariate extremes. Biometrika 107, 513–532.CrossRefGoogle Scholar