Bernoulli

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

A new representation for multivariate tail probabilities

J.L. Wadsworth and J.A. Tawn

Full-text: Open access

Abstract

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.

Article information

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

Dates
First available in Project Euclid: 3 December 2013

Permanent link to this document
https://projecteuclid.org/euclid.bj/1386078617

Digital Object Identifier
doi:10.3150/12-BEJ471

Mathematical Reviews number (MathSciNet)
MR3160568

Zentralblatt MATH identifier
1284.60107

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

Citation

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. https://projecteuclid.org/euclid.bj/1386078617


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