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Hidden regular variation and the rank transform

Part of: Multivariate analysis Distribution theory - Probability Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Janet Heffernan  and
Sidney Resnick
Janet Heffernan*
Affiliation:
Lancaster University
Sidney Resnick*
Affiliation:
Cornell University
*
∗ Postal address: Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, UK. Email address: j.heffernan@lancaster.ac.uk
∗∗ Postal address: School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY 14853, USA. Email address: sir1@cornell.edu
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Abstract

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Random vectors in the positive orthant whose distributions possess hidden regular variation are a subclass of those whose distributions are multivariate regularly varying with asymptotic independence. The concept is an elaboration of the coefficient of tail dependence of Ledford and Tawn. We show that the rank transform that brings unequal marginals to the standard case also preserves the hidden regular variation. We discuss applications of the results to two examples, one involving flood risk and the other Internet data.

Keywords

Heavy tailregular variationPareto tailcoefficient of tail dependencehidden regular variationrank transformasymptotic independenceInternet trafficflood risk

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 62H05: Characterization and structure theory 62H20: Measures of association (correlation, canonical correlation, etc.) 60E05: Distributions
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 2 , June 2005 , pp. 393 - 414
Copyright
Copyright © Applied Probability Trust 2005 

References

Beirlant, J. and Vandewalle, B. (2002). Some comments on the estimation of a dependence index in bivariate extreme value statistics. Statist. Prob. Lett. 60, 265–278.Google Scholar
Beirlant, J., Vynckier, P. and Teugels, J. (1996). Tail index estimation, Pareto quantile plots, and regression diagnostics. J. Amer. Statist. Assoc. 91, 1659–1667.Google Scholar
Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
Bruun, J. T. and Tawn, J. A. (1998). Comparison of approaches for estimating the probability of coastal flooding. J. R. Statist. Soc. C 47, 405–423.Google Scholar
Campos, F. H. et al. (2005). Extremal dependence: Internet traffic applications. Stoch. Models 21, 1–35.Google Scholar
Coles, S. G., Heffernan, J. E. and Tawn, J. A. (1999). Dependence measures for extreme value analyses. Extremes 2, 339–365.Google Scholar
De Haan, L. and de Ronde, J. (1998). Sea and wind: multivariate extremes at work. Extremes 1, 7–46.Google Scholar
De Haan, L. and Omey, E. (1984). Integrals and derivatives of regularly varying functions in R d and domains of attraction of stable distributions. II. Stoch. Process. Appl. 16, 157–170.Google Scholar
Draisma, G., Drees, H., Ferreira, A. and de Haan, L. (2004). Bivariate tail estimation: dependence in asymptotic independence. Bernoulli 10, 251–280.CrossRefGoogle Scholar
Drees, H., de Haan, L. and Resnick, S. I. (2000). How to make a Hill plot. Ann. Statist. 28, 254–274.CrossRefGoogle Scholar
Drees, H., Ferreira, A. and de Haan, L. (2004). On maximum likelihood estimation of the extreme value index. Ann. Appl. Prob. 14, 1179–1201.Google Scholar
Greenwood, P. and Resnick, S. (1979). A bivariate stable characterization and domains of attraction. J. Multivariate Anal. 9, 206–221.Google Scholar
Hand, W. H. (2002). Numerical weather prediction. A historical study of rainfall events in the 20th century. Forecast. Tech. Rep. 384, UK Met Office. Available at http://www.metoffice.com/research/nwp/publications/papers/technical_reports/2002/FRTR384/FRTR384.pdf.Google Scholar
Heffernan, J. E. (2000). A directory of coefficients of tail dependence. Extremes 3, 279–290.CrossRefGoogle Scholar
Huang, X. (1992). Statistics of bivariate extreme values. , Erasmus University Rotterdam.Google Scholar
Hult, H. and Lindskog, F. (2002). Multivariate extremes, aggregation and dependence in elliptical distributions. Adv. Appl. Prob. 34, 587–608.Google Scholar
Kratz, M. and Resnick, S. I. (1996). The QQ-estimator and heavy tails. Commun. Statist. Stoch. Models 12, 699–724.Google Scholar
Ledford, A. W. and Tawn, J. A. (1996). Statistics for near independence in multivariate extreme values. Biometrika 83, 169–187.Google Scholar
Ledford, A. W. and Tawn, J. A. (1997). Modelling dependence within Joint tail regions. J. R. Statist. Soc. B 59, 475–499.Google Scholar
Maulik, K. and Resnick, S. I. (2005). Characterizations and examples of hidden regular variation. Extremes 7, 31–67.Google Scholar
Maulik, K., Resnick, S. I. and Rootzén, H. (2002). Asymptotic independence and a network traffic model. J. Appl. Prob. 39, 671–699.Google Scholar
Nadarajah, S., Anderson, C. W. and Tawn, J. A. (1998). Ordered multivariate extremes. J. R. Statist. Soc. B 60, 473–496.Google Scholar
Peng, L. (1999). Estimation of the coefficient of tail dependence in bivariate extremes. Statist. Prob. Lett. 43, 399–409.Google Scholar
Poon, S.-H., Rockinger, M. and Tawn, J. (2003). Modelling extreme-value dependence in international stock markets. Statistica Sinica 13, 929–953.Google Scholar
Resnick, S. I. (1986). Point processes, regular variation and weak convergence. Adv. Appl. Prob. 18, 66–138.Google Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.Google Scholar
Resnick, S. (2002). Hidden regular variation, second order regular variation and asymptotic independence. Extremes 5, 303–336.CrossRefGoogle Scholar
Resnick, S. (2003). Modeling data networks. In Exteme Values in Finance, Telecommunications, and the Environment, eds Finkenstadt, B. and Rootzen, H., Chapman and Hall, London, pp. 287–372.Google Scholar
Resnick, S. (2004). The extremal dependence measure and asymptotic independence. Stoch. Models 20, 205–227.Google Scholar
Resnick, S. and Stărică, C. (1997). Smoothing the Hill estimator. Adv. Appl. Prob. 29, 271–293.CrossRefGoogle Scholar
Stărică, C. (1999). Multivariate extremes for models with constant conditional correlations. J. Empirical Finance 6, 515–553.Google Scholar