A new representation for multivariate tail probabilities

By Jennifer L. Wadsworth, Jonathan A. Tawn. 2013

64 citations recorded by the source corpus.

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.

Subfields: Economics and Econometrics, Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/254c366396e268320dee1653661bf73b7334fea2

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Citations in the corpus

Modeling asymptotically independent spatial extremes based on Laplace random fields (2015, methodology): “We follow common practice in spatial extreme value modeling and use the empirical distribution below the threshold by assuming dense observations in the bulk region of the distribution (Coles and Tawn, 1991; Wadsworth and Tawn, 2012, 2013).”
Determining the dependence structure of multivariate extremes (2018, background): “If χC = 0, the variables in XC cannot all take their largest values together, although it is possible that for any C ⊂ C, χC > 0, see for example Hua and Joe (2011) or Wadsworth and Tawn (2013).”
Modelling across extremal dependence classes (2014, background): “This special case of the set-up of Wadsworth and Tawn (2013) is satisfied by the Morgenstern, inverted extreme value, Ali–Mikhail–Haq, and Pareto copulas, amongst others; see Heffernan (2000) for a summary of their extremal dependence properties.”