样本最大值的控制与相关极值相关测度
Domination of Sample Maxima and Related Extremal Dependence Measures
摘要：
对于一个给定的$d$维分布函数（df）$H$，我们引入一类相关测度$ μ(H,Q) = - \mathbb{E}\{ \ln H(Z_1, \ldots, Z_d)\},$，其中随机向量$(Z_1, \ldots, Z_d)$具有与$H$具有相同边际df的df $Q$.如果这两个$H$和$Q$是最大稳定的DF的，我们表明，对于DF $F$在最大域的吸引力的$H$，这种依赖措施解释了极值依赖表现出的$F$。证明了$μ(H,Q)$是在$Q$的最大吸引域中，$F$随机样本的最大值被具有df的随机向量边缘支配的概率的极限.我们展示了一个类似的结果，完全支配的样本最大值，这导致另一种措施的依赖表示为$λ(Q,H)$。在文献$λ(H,H)$与$H$一个最大稳定的DF已被研究的背景下，记录，多个最大值，伴随的顺序统计量和并发概率。事实证明，$μ(H,Q)$和$λ(Q,H)$密切相关。如果$H$是最大稳定的，我们得到有用的表示为$μ(H,Q)$和$λ(Q,H)$。我们的应用程序包括等价条件$H$是一个产品DF和$F$有渐近独立的组件。
英文摘要：
For a given $d$-dimensional distribution function (df) $H$ we introduce the class of dependence measures $ μ(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 $μ(H,Q)$ 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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