Domination of Sample Maxima and Related Extremal Dependence Measures

For a given $d$-dimensional distribution function (df) $H$ we introduce the class of dependence measures $ \mu(H,Q) = - \mathbb{E}\{ \ln H(Z_1, \ldots, Z_d)\},$ where the random vector $(Z_1, \ldots, Z_d)$ has df $Q$ which has the same marginal df's as $H$. If both $H$ and $Q$ are max-stable df's, we show that for a df $F$ in the max-domain of attraction of $H$, this dependence measure explains the extremal dependence exhibited by $F$. Moreover we prove that $\mu(H,Q)$ is the limit of the probability that the maxima of a random sample from $F$ is marginally dominated by some random vector with df in the max-domain of attraction of $Q$. We show a similar result for the complete domination of the sample maxima which leads to another measure of dependence denoted by $\lambda(Q,H)$. In the literature $\lambda(H,H)$ with $H$ a max-stable df has been studied in the context of records, multiple maxima, concomitants of order statistics and concurrence probabilities. It turns out that both $\mu(H,Q)$ and $\lambda(Q,H)$ are closely related. If $H$ is max-stable we derive useful representations for both $\mu(H,Q)$ and $\lambda(Q,H)$. Our applications include equivalent conditions for $H$ to be a product df and $F$ to have asymptotically independent components.