Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - A new representation for multivariate tail probabilities
AU - Wadsworth, Jennifer
AU - Tawn, Jon
PY - 2013
Y1 - 2013
N2 - 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.
AB - 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.
KW - asymptotic independence
KW - coefficient of tail dependence
KW - multivariate extreme value theory
KW - Pickands’ dependence function
KW - regular variation
U2 - 10.3150/12-BEJ471
DO - 10.3150/12-BEJ471
M3 - Journal article
VL - 19
SP - 2689
EP - 2714
JO - Bernoulli
JF - Bernoulli
SN - 1350-7265
IS - 5B
ER -