• Bernoulli
  • Volume 19, Number 5B (2013), 2689-2714.

A new representation for multivariate tail probabilities

J.L. Wadsworth and J.A. Tawn

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Existing theory for multivariate extreme values focuses upon characterizations of the distributional tails when all components of a random vector, standardized to identical margins, grow at the same rate. In this paper, we consider the effect of allowing the components to grow at different rates, and characterize the link between these marginal growth rates and the multivariate tail probability decay rate. Our approach leads to a whole class of univariate regular variation conditions, in place of the single but multivariate regular variation conditions that underpin the current theories. These conditions are indexed by a homogeneous function and an angular dependence function, which, for asymptotically independent random vectors, mirror the role played by the exponent measure and Pickands’ dependence function in classical multivariate extremes. We additionally offer an inferential approach to joint survivor probability estimation. The key feature of our methodology is that extreme set probabilities can be estimated by extrapolating upon rays emanating from the origin when the margins of the variables are exponential. This offers an appreciable improvement over existing techniques where extrapolation in exponential margins is upon lines parallel to the diagonal.

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Bernoulli, Volume 19, Number 5B (2013), 2689-2714.

First available in Project Euclid: 3 December 2013

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asymptotic independence coefficient of tail dependence multivariate extreme value theory Pickands’ dependence function regular variation


Wadsworth, J.L.; Tawn, J.A. A new representation for multivariate tail probabilities. Bernoulli 19 (2013), no. 5B, 2689--2714. doi:10.3150/12-BEJ471.

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  • [1] Beirlant, J., Goegebeur, Y., Teugels, J. and Segers, J. (2004). Statistics of Extremes: Theory and Applications. Wiley Series in Probability and Statistics. Chichester: Wiley.
  • [2] Coles, S.G. and Tawn, J.A. (1991). Modelling extreme multivariate events. J. R. Stat. Soc. Ser. B Stat. Methodol. 53 377–392.
  • [3] de Haan, L. and de Ronde, J. (1998). Sea and wind: Multivariate extremes at work. Extremes 1 7–45.
  • [4] de Haan, L. and Resnick, S. (1998). On asymptotic normality of the Hill estimator. Comm. Statist. Stochastic Models 14 849–866.
  • [5] Haeusler, E. and Teugels, J.L. (1985). On asymptotic normality of Hill’s estimator for the exponent of regular variation. Ann. Statist. 13 743–756.
  • [6] Hall, P. and Tajvidi, N. (2000). Distribution and dependence-function estimation for bivariate extreme-value distributions. Bernoulli 6 835–844.
  • [7] Hashorva, E. and Hüsler, J. (2003). On multivariate Gaussian tails. Ann. Inst. Statist. Math. 55 507–522.
  • [8] Heffernan, J. and Resnick, S. (2005). Hidden regular variation and the rank transform. Adv. in Appl. Probab. 37 393–414.
  • [9] Heffernan, J.E. and Resnick, S.I. (2007). Limit laws for random vectors with an extreme component. Ann. Appl. Probab. 17 537–571.
  • [10] Heffernan, J.E. and Tawn, J.A. (2004). A conditional approach for multivariate extreme values (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 66 497–546.
  • [11] Hill, B.M. (1975). A simple general approach to inference about the tail of a distribution. Ann. Statist. 3 1163–1174.
  • [12] Joe, H. (1997). Multivariate Models and Dependence Concepts. Monographs on Statistics and Applied Probability 73. London: Chapman & Hall.
  • [13] Ledford, A.W. and Tawn, J.A. (1996). Statistics for near independence in multivariate extreme values. Biometrika 83 169–187.
  • [14] Ledford, A.W. and Tawn, J.A. (1997). Modelling dependence within joint tail regions. J. R. Stat. Soc. Ser. B Stat. Methodol. 59 475–499.
  • [15] Maulik, K. and Resnick, S. (2004). Characterizations and examples of hidden regular variation. Extremes 7 31–67.
  • [16] Niculescu, C.P. and Persson, L.E. (2006). Convex Functions and Their Applications: A Contemporary Approach. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC 23. New York: Springer.
  • [17] Peng, L. and Qi, Y. (2004). Discussion of “A conditional approach for multivariate extreme values,” by J. E. Heffernan and J. A. Tawn. J. R. Stat. Soc. Ser. B Stat. Methodol. 66 541–542.
  • [18] Pickands, J. (1981). Multivariate extreme value distributions. In Bulletin of the International Statistical Institute: Proceedings of the 43rd Session (Buenos Aires) 859–878. Voorburg, Netherlands: ISI.
  • [19] Ramos, A. and Ledford, A. (2009). A new class of models for bivariate joint tails. J. R. Stat. Soc. Ser. B Stat. Methodol. 71 219–241.
  • [20] Resnick, S. (2002). Hidden regular variation, second order regular variation and asymptotic independence. Extremes 5 303–336.
  • [21] Resnick, S.I. (1987). Extreme Values, Regular Variation, and Point Processes. Applied Probability. A Series of the Applied Probability Trust 4. New York: Springer.
  • [22] Resnick, S.I. (2007). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer Series in Operations Research and Financial Engineering. New York: Springer.
  • [23] Ruben, H. (1964). An asymptotic expansion for the multivariate normal distribution and Mills’ ratio. J. Res. Nat. Bur. Standards Sect. B 68B 3–11.
  • [24] Savage, I.R. (1962). Mills’ ratio for multivariate normal distributions. Journal of Research of the National Bureau of Standards B 66B 93–96.
  • [25] Tawn, J.A. (1988). Bivariate extreme value theory: Models and estimation. Biometrika 75 397–415.
  • [26] Wadsworth, J.L. (2012). Models for penultimate extreme values. Ph.D. thesis, Lancaster Univ.