Domination of Sample Maxima and Related Extremal Dependence Measures
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For a given d-dimensional distribution function (df) H we introduce the class of dependence measures μ(H,Q) = - E{ H(Z_1, ..., Z_d)}, where the random vector (Z_1, ..., 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 μ(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 λ(Q,H). In the literature λ(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 μ(H,Q) and λ(Q,H) are closely related. If H is max-stable we derive useful representations for both μ(H,Q) and λ(Q,H). Our applications include equivalent conditions for H to be a product df and F to have asymptotically independent components.
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