Inequalities for the extremal coefficients of multivariate extreme value distributions (Q1424669)

Let \(A\) be a finite index set and \(Z_i,i\in A\), be a random vector with a multivariate extreme value distribution with identical univariate margins. The Smith extremal coefficient \(\theta_A\) is defined by \(\Pr(\max_{j\in A} Z_j<x)=(\Pr(Z_j<x))^{\theta_A}\), \(\forall x\in R\). In terms of Pickands measure \(H\) it can be defined as \(\theta_A=\int \max_{j\in A} x_j d H(x_1,\dots,x_m). \) The authors consider some characterizations and inequalities for \(\theta_A\). E.g. \( \theta_X\leq \min_{A,B\subseteq X, A\cup B=X} (\theta_A+\theta_B-\theta_{A\cap B}).\) Special attention is paid to the four-dimensional case.