Open Access
November 2013 A new representation for multivariate tail probabilities
J.L. Wadsworth, J.A. Tawn
Bernoulli 19(5B): 2689-2714 (November 2013). DOI: 10.3150/12-BEJ471


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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J.L. Wadsworth. J.A. Tawn. "A new representation for multivariate tail probabilities." Bernoulli 19 (5B) 2689 - 2714, November 2013.


Published: November 2013
First available in Project Euclid: 3 December 2013

zbMATH: 1284.60107
MathSciNet: MR3160568
Digital Object Identifier: 10.3150/12-BEJ471

Keywords: Asymptotic independence , coefficient of tail dependence , multivariate extreme value theory , Pickands’ dependence function , regular variation

Rights: Copyright © 2013 Bernoulli Society for Mathematical Statistics and Probability

Vol.19 • No. 5B • November 2013
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