Abstract
The dependence structure of a multivariate extreme value (MEV) distribution is characterized by its spectral measure. In this paper, we investigate the interconnections between the supermodular ordering of two d-dimensional MEV distributions and the convex ordering of their spectral measures. The main result reveals some insightful understanding of the dependence structures of MEV distributions. More precisely, let G and G ∗ be two MEV distributions on R d with the corresponding univariate margins equal, and let S and S ∗ be their respective spectral measures with respect to the ℓ 1-norm ∥⋅∥. Suppose that W and W ∗ are two random vectors taking values on \({\Theta }=\{\theta \in \Re _{+}^{d}: \|\theta \|=1\}\) according to the probability laws S/d and S ∗/d, respectively. If W is smaller than W ∗ in the convex order, then G ∗ is smaller than G in the supermodular order. Several applications of the main result are also presented.
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Mao, T., Hu, T. Relations between the spectral measures and dependence of MEV distributions. Extremes 18, 65–84 (2015). https://doi.org/10.1007/s10687-014-0203-z
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DOI: https://doi.org/10.1007/s10687-014-0203-z
Keywords
- Multivariate extreme value distribution
- Multivariate extreme value copula
- Spectral measure
- Simplex
- Convex order; supermodular order
- Convex transfer