Abstract
A sub-model of multivariate regular variation called hidden regular variation facilitates more accurate estimation of joint tail probabilities in the presence of asymptotic independence. A related concept called hidden domain of attraction can sometimes offer similar estimation assistance in circumstances where hidden regular variation is absent. Examples and discussion illustrate strengths and limitations of this concept. We outline estimation techniques where applicable.
Similar content being viewed by others
References
Balkema, A.A., de Haan, L.: On R. von Mises’ condition for the domain of attraction of \(\exp (-e^{-x})^{1}\). Ann. Math. Stat. 43, 1352–1354 (1972). ISSN 0003-4851
Balkema, A.A., de Haan, L.: Limit distributions for order statistics. I. Teor. Verojatnost. i Primenen. 23(1), 80–96 (1978a). ISSN 0040-361x
Balkema, A.A., de Haan, L.: Limit distributions for order statistics. II. Teor. Verojatnost. i Primenen. 23(2), 358–375 (1978b). ISSN 0040-361x
Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley, New York (1999). ISBN 0-471-19745-9. A Wiley-Interscience Publication
Bruun, J.T., Tawn, J.A.: Comparison of approaches for estimating the probability of coastal flooding. J. R. Stat. Soc. Ser. C Appl. Stat. 47(3), 405–423 (1998)
Das, B., Resnick, S.I.: Conditioning on an extreme component: model consistency with regular variation on cones. Bernoulli 17(1), 226–252 (2011a). doi:10.3150/10-BEJ271. ISSN 1350-7265
Das, B., Resnick, S.I.: Detecting a conditional extreme value model. Extremes 14(1), 29–61 (2011b)
Das, B., Mitra, A., Resnick, S.: Living on the multi-dimensional edge: seeking hidden risks using regular variation. Adv. Appl. Probab. 45(1), 1–25 (2013). arXiv:1108.5560
de Haan, L.: On Regular Variation and its Application to the Weak Convergence of Sample Extremes. Mathematisch Centrum, Amsterdam (1970)
de Haan, L.: A characterization of multidimensional extreme-value distributions. Sankhyā Ser A 40(1), 85–88 (1978). ISSN 0581-572X
de Haan, L., de Ronde, J.: Sea and wind: multivariate extremes at work. Extremes 1(1), 7–46 (1998)
de Haan, L., Ferreira, A.: Extreme, Value Theory: An Introduction. Springer, New York (2006)
Heffernan, J.E., Resnick, S.I.: Hidden regular variation and the rank transform. Adv. Appl. Probab. 37(2), 393–414 (2005)
Heffernan, J.E., Resnick, S.I.: Limit laws for random vectors with an extreme component. Ann. Appl. Probab. 17(2), 537–571 (2007). doi:10.1214/105051606000000835. ISSN 1050-5164
Heffernan, J.E., Tawn, J.A.: A conditional approach for multivariate extreme values (with discussion). JRSS B 66(3), 497–546 (2004)
Huang, X.: Statistics of Bivariate Extreme Values. Ph.D. thesis, Tinbergen Institute Research Series 22. Erasmus University Rotterdam, Postbus 1735, 3000DR, Rotterdam, The Netherlands (1992)
Ledford, A.W., Tawn, J.A.: Statistics for near independence in multivariate extreme values. Biometrika 83(1), 169–187 (1996). ISSN 0006-3444
Ledford, A.W., Tawn, J.A.: Modelling dependence within joint tail regions. J. R. Stat. Soc. Ser. B 59(2), 475–499 (1997). ISSN 0035-9246
Ledford, A.W., Tawn, J.A.: Concomitant tail behaviour for extremes. Adv. Appl. Probab. 30(1), 197–215 (1998). ISSN 0001-8678
Maulik, K., Resnick, S.I.: Characterizations and examples of hidden regular variation. Extremes 7(1), 31–67 (2005)
Mitra, A., Resnick, S.I.: Hidden regular variation: detection and estimation (2010). arXiv:1001.5058
Mitra, A., Resnick, S.I.: Hidden regular variation and detection of hidden risks. Stoch. Models 27(4), 591–614 (2011)
Poon, S.-H., Rockinger, M., Tawn, J.: Modelling extreme-value dependence in international stock markets. Stat. Sin. 13(4), 929–953 (2003). ISSN 1017-0405. Statistical applications in financial econometrics
Resnick, S.I.: A Probability Path. Birkhäuser, Boston (1999)
Resnick, S.I.: Hidden regular variation, second order regular variation and asymptotic independence. Extremes 5(4), 303–336 (2003). ISSN 1386-1999
Resnick, S.I.: Heavy Tail Phenomena: Probabilistic and Statistical Modeling. Springer Series in Operations Research and Financial Engineering. Springer, New York (2007). ISBN 0-387-24272-4
Resnick, S.I.: Extreme Values, Regular Variation and Point Processes. Springer, New York (2008). ISBN 978-0-387-75952-4. Reprint of the 1987 original
Smirnov, N.V.: Limit distributions for the terms of a variational series. Trudy Mat. Inst. Steklov. 25, 60 (1949). ISSN 0371-9685
Smith, R.L.: Statistics of extremes, with applications in environment, insurance and finance. In: Finkenstadt, B., Rootzén, H. (eds.) SemStat: Seminaire Europeen de Statistique, Extreme Values in Finance, Telecommunications, and the Environment, pp. 1–78. Chapman-Hall, London (2003)
Author information
Authors and Affiliations
Corresponding author
Additional information
S. Resnick and A. Mitra were partially supported by ARO Contract W911NF-10-1-0289 at Cornell University. S. Resnick also received support from NSA Grant H98230-11-1-0193.
Rights and permissions
About this article
Cite this article
Mitra, A., Resnick, S.I. Modeling multiple risks: hidden domain of attraction. Extremes 16, 507–538 (2013). https://doi.org/10.1007/s10687-013-0171-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10687-013-0171-8