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Hidden regular variation and the rank transform. (English) Zbl 1073.60057
Summary: 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 A. W. Ledford and J. A. Tawn [Biometrika 83, 169–187 (1996; Zbl 0865.62040) and J. R. Stat. Soc., Ser. B 59, 475–499 (1997; Zbl 0886.62063)]. 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.

MSC:
60G70 Extreme value theory; extremal stochastic processes
62H05 Characterization and structure theory for multivariate probability distributions; copulas
62H20 Measures of association (correlation, canonical correlation, etc.)
60E05 Probability distributions: general theory
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[1] 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. · Zbl 1092.62544 · doi:10.1016/S0167-7152(02)00281-X
[2] 1996beirlant:vynckier:teugels:1996 Beirlant, J., Vynckier, P. and Teugels, J. (1996). Tail index estimation, Pareto quantile plots, and regression diagnostics. J. Amer. Statist. Assoc. 91, 1659–1667. · Zbl 0881.62077 · doi:10.2307/2291593
[3] Billingsley, P. (1968). Convergence of Probability Measures . John Wiley, New York. · Zbl 0172.21201
[4] 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. · Zbl 0905.62123 · doi:10.1111/1467-9876.00118
[5] 2005campos:marron:resnick:jaffay:2002 Campos, F. H. et al. (2005). Extremal dependence: Internet traffic applications. Stoch. Models 21 , 1–35. · Zbl 1061.62077 · doi:10.1081/STM-200046446
[6] 1999coles:heffernan:tawn:1999 Coles, S. G., Heffernan, J. E. and Tawn, J. A. (1999). Dependence measures for extreme value analyses. Extremes 2, 339–365. · Zbl 0972.62030
[7] De Haan, L. and de Ronde, J. (1998). Sea and wind: multivariate extremes at work. Extremes 1, 7–46. · Zbl 0921.62144 · doi:10.1023/A:1009909800311
[8] De Haan, L. and Omey, E. (1984). Integrals and derivatives of regularly varying functions in \(\mathbbR^d\) and domains of attraction of stable distributions. II. Stoch. Process. Appl. 16, 157–170. · Zbl 0528.60021 · doi:10.1016/0304-4149(84)90016-4
[9] 2004draisma:drees:ferreira:dehaan:2001 Draisma, G., Drees, H., Ferreira, A. and de Haan, L. (2004). Bivariate tail estimation: dependence in asymptotic independence. Bernoulli 10, 251–280. · Zbl 1058.62043 · doi:10.3150/bj/1082380219
[10] 2000drees:dehaan:resnick:2000 Drees, H., de Haan, L. and Resnick, S. I. (2000). How to make a Hill plot. Ann. Statist. 28, 254–274. · Zbl 1106.62333 · doi:10.1214/aos/1016120372 · euclid:aos/1016120372
[11] 2004drees:ferreira:dehaan:2004 Drees, H., Ferreira, A. and de Haan, L. (2004). On maximum likelihood estimation of the extreme value index. Ann. Appl. Prob. 14, 1179–1201. · Zbl 1102.62051 · doi:10.1214/105051604000000279
[12] Greenwood, P. and Resnick, S. (1979). A bivariate stable characterization and domains of attraction. J. Multivariate Anal. 9, 206–221. · Zbl 0409.62038 · doi:10.1016/0047-259X(79)90079-4
[13] 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.
[14] Heffernan, J. E. (2000). A directory of coefficients of tail dependence. Extremes 3, 279–290. · Zbl 0979.62040 · doi:10.1023/A:1011459127975
[15] Huang, X. (1992). Statistics of bivariate extreme values. Doctoral Thesis, Erasmus University Rotterdam.
[16] Hult, H. and Lindskog, F. (2002). Multivariate extremes, aggregation and dependence in elliptical distributions. Adv. Appl. Prob. 34, 587–608. · Zbl 1023.60021 · doi:10.1239/aap/1033662167
[17] Kratz, M. and Resnick, S. I. (1996). The QQ-estimator and heavy tails. Commun. Statist. Stoch. Models 12, 699–724. · Zbl 0887.62025 · doi:10.1080/15326349608807407
[18] Ledford, A. W. and Tawn, J. A. (1996). Statistics for near independence in multivariate extreme values. Biometrika 83, 169–187. · Zbl 0865.62040 · doi:10.1093/biomet/83.1.169 · www3.oup.co.uk
[19] Ledford, A. W. and Tawn, J. A. (1997). Modelling dependence within joint tail regions. J. R. Statist. Soc. B 59, 475–499. · Zbl 0886.62063 · doi:10.1111/1467-9868.00080
[20] Maulik, K. and Resnick, S. I. (2005). Characterizations and examples of hidden regular variation. Extremes 7 , 31–67. · Zbl 1088.62066 · doi:10.1007/s10687-004-4728-4
[21] 2002maulik:resnick:rootzen:2002 Maulik, K., Resnick, S. I. and Rootzén, H. (2002). Asymptotic independence and a network traffic model. J. Appl. Prob. 39, 671–699. · Zbl 1090.90017 · doi:10.1239/jap/1037816012
[22] 1998nadarajah:anderson:tawn:1998 Nadarajah, S., Anderson, C. W. and Tawn, J. A. (1998). Ordered multivariate extremes. J. R. Statist. Soc. B 60, 473–496. · Zbl 0910.62054 · doi:10.1111/1467-9868.00136
[23] Peng, L. (1999). Estimation of the coefficient of tail dependence in bivariate extremes. Statist. Prob. Lett. 43, 399–409. · Zbl 0958.62049 · doi:10.1016/S0167-7152(98)00280-6
[24] 2003poon:rockinger:tawn:2003 Poon, S.-H., Rockinger, M. and Tawn, J. (2003). Modelling extreme-value dependence in international stock markets. Statistica Sinica 13, 929–953. · Zbl 1034.62106
[25] Resnick, S. I. (1986). Point processes, regular variation and weak convergence. Adv. Appl. Prob. 18, 66–138. · Zbl 0597.60048 · doi:10.2307/1427239
[26] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes . Springer, New York. · Zbl 0633.60001
[27] Resnick, S. (2002). Hidden regular variation, second order regular variation and asymptotic independence. Extremes 5, 303–336. · Zbl 1035.60053 · doi:10.1023/A:1025148622954
[28] Resnick, S. (2003). Modeling data networks. In Exteme Values in Finance, Telecommunications, and the Environment , eds B. Finkenstadt and H. Rootzen, Chapman and Hall, London, pp. 287–372.
[29] Resnick, S. (2004). The extremal dependence measure and asymptotic independence. Stoch. Models 20, 205–227. · Zbl 1054.62063 · doi:10.1081/STM-120034129
[30] Resnick, S. and Stărică, C. (1997). Smoothing the Hill estimator. Adv. Appl. Prob. 29, 271–293. · Zbl 0873.60021 · doi:10.2307/1427870
[31] Stărică, C. (1999). Multivariate extremes for models with constant conditional correlations. J. Empirical Finance 6, 515–553.
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