A new representation for multivariate tail probabilities

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.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ471 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Approximation and estimation of very small probabilities of multivariate extreme events

“…Defining a function κ by κ(a) := inf I (A a ) for every a ∈ [0, ∞) m , Eq. 4.7 becomes identical to an extension of RTD recently introduced in Wadsworth and Tawn (2013). Wadsworth and Tawn (2013) proposed this assumption to close the possible gap between (4.1) and the regularity of the marginal tails.…”

“…4.7 becomes identical to an extension of RTD recently introduced in Wadsworth and Tawn (2013). Wadsworth and Tawn (2013) proposed this assumption to close the possible gap between (4.1) and the regularity of the marginal tails. It is curious that this condition, requiring the existence of separate limits of the survival function along chosen paths, is derivable from the simple LDP (3.1).…”

“…By Example 1(a,b) in Table 1 of Wadsworth and Tawn (2013), κ(x) = I (x) if min(x 1 /x 2 , x 2 /x 2 ) > ρ 2 or if ρ < 0 and min(x 1 , x 2 ) > 0, and κ(x) = max(x 1 , x 2 ) for all other x ∈ [0, ∞) 2 . The left and middle panels of Fig.…”

“…4.7 with inf I (A a ) replaced by κ(a) to the apparently new limit relation (4.6) is not trivial. Another generalisation, proposed in Wadsworth and Tawn (2013), is 8) derived in Section 3.3 of Wadsworth and Tawn (2013) for the bivariate case under the assumption that κ is differentiable and a ∈ [0, ∞) m \ {0} satisfies ∂κ(a)/∂a j > 0 for j = 1, .., m. As noted in Wadsworth and Tawn (2013), a in Eq. 4.8 would have to be chosen in an application.…”

“…Ledford and Tawn (1996, 1997, Peng (1999), Resnick (2002), Draisma et al (2004) and Heffernan and Resnick (2005). This model was recently extended in Wadsworth and Tawn (2013). Another approach, based on conditional limits, was proposed in Heffernan and Tawn (2004) and Heffernan and Resnick (2007).…”