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Living on the multidimensional edge: Seeking hidden risks using regular variation. (English) Zbl 1276.60041
The authors consider a nonnegative random vector which they interpret as a risk vector. The distribution of the vector is supposed to have multivariate regular variation that is an asymptotic relation which is called \(M^*\)-convergence. The notion of multivariate regular variation supplies tail equivalence of the marginal components, so it has to be modified for applications. But its advantage is that it allows the limit measure to be used for the approximation of the tail probabilities. Such approximations of tail probabilities are sensitive to degeneracies in the limit measure. Other advantages and disadvantages of this approach comparing to hidden regular variation (HRV) are analyzed. The \(M^*\)-convergence is reviewed and the reasons to abandon the standard practice of defining regular variation through vague convergence on a compactification of \(\mathbb{R}^d\) are discussed. A nonstandard regular variation is considered from the point of view of regular variation on the sequence of cones. Nonstandard regular variation is essential in practice since it is not natural to assume in applications that all marginals in a multivariate model are tail equivalent. In the conclusion, it is discussed how to fit the model of HRV on cones to data as well as estimation techniques of tail probabilities using the model introduced in the paper.

60F99 Limit theorems in probability theory
62G32 Statistics of extreme values; tail inference
60G70 Extreme value theory; extremal stochastic processes
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[1] Billingsley, P. (1999). Convergence of Probability Measures , 2nd edn. John Wiley, New York. · Zbl 0944.60003
[2] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation (Encyclopedia Math. Appl. 27 ). Cambridge University Press. · Zbl 0667.26003
[3] Cai, J.-J., Einmahl, J. H. J. and de Haan, L. (2011). Estimation of extreme risk regions under multivariate regular variation. Ann. Statist. 39 , 1803-1826. · Zbl 1221.62075
[4] Das, B. and Resnick, S. (2011). Conditioning on an extreme component: model consistency with regular variation on cones. Bernoulli 17 , 226-252. · Zbl 1284.60103
[5] De Haan, L. and Ferreira, A. (2006). Extreme Value Theory . Springer, New York. · Zbl 1101.62002
[6] De Haan, L. and Resnick, S. (1993). Estimating the limit distribution of multivariate extremes. Commun. Statist. Stoch. Models 9 , 275-309. · Zbl 0777.62036
[7] Heffernan, J. and Resnick, S. (2005). Hidden regular variation and the rank transform. Adv. Appl. Prob. 37 , 393-414. · Zbl 1073.60057
[8] Heffernan, J. E. and Resnick, S. I. (2007). Limit laws for random vectors with an extreme component. Ann. Appl. Prob. 17 , 537-571. · Zbl 1125.60049
[9] Heffernan, J. E. and Tawn, J. A. (2004). A conditional approach for multivariate extreme values (with discussion). J. R. Statist. Soc. B 66 , 497-546. · Zbl 1046.62051
[10] Hult, H. and Lindskog, F. (2006). Regular variation for measures on metric spaces. Publ. Inst. Math. (Beograd) (N.S.) 80 (94), 121-140. · Zbl 1164.28005
[11] Joe, H. and Li, H. (2011). Tail risk of multivariate regular variation. Methodology Comput. Appl. Prob. 13 , 671-693. · Zbl 1239.62060
[12] Kallenberg, O. (1983). Random Measures , 3rd edn. Akademie, Berlin. · Zbl 0544.60053
[13] Ledford, A. W. and Tawn, J. A. (1996). Statistics for near independence in multivariate extreme values. Biometrika 83 , 169-187. · Zbl 0865.62040
[14] Ledford, A. W. and Tawn, J. A. (1998). Concomitant tail behaviour for extremes. Adv. Appl. Prob. 30 , 197-215. · Zbl 0905.60034
[15] Maulik, K. and Resnick, S. (2005). Characterizations and examples of hidden regular variation. Extremes 7 , 31-67. · Zbl 1088.62066
[16] Mitra, A. and Resnick, S. I. (2010). Hidden regular variation: detection and estimation. Preprint. Available at http://arxiv.org/abs/1001.5058v2.
[17] Mitra, A. and Resnick, S. I. (2011). Hidden regular variation and detection of hidden risks. Stoch. Models 27 , 591-614. · Zbl 1230.91080
[18] Molchanov, I. (2005). Theory of Random Sets . Springer, London. · Zbl 1109.60001
[19] Prohorov, Y. V. (1956). Convergence of random processes and limit theorems in probability theory. Teor. Verojat. Primen. 1 , 177-238. · Zbl 0075.29001
[20] Resnick, S. (2002). Hidden regular variation, second order regular variation and asymptotic independence. Extremes 5 , 303-336. · Zbl 1035.60053
[21] Resnick, S. (2007). Heavy Tail Phenomena: Probabilistic and Statistical Modeling . Springer, New York. · Zbl 1152.62029
[22] Resnick, S. (2008). Extreme Values, Regular Variation and Point Processes . Springer, New York. · Zbl 1136.60004
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