从Meta密度看样品云极限形状的灵敏度
Sensitivity of the limit shape of sample clouds from meta densities
摘要：
研究了一类轻尾多元概率分布。这些是通过从重尾原始分布的利润率的变换获得的。这类是在J. Multivariate Anal. 101（2010）1738-1754）。如图所示，对于轻尾Meta分布，适当缩放的样本云收敛到一个确定性集合上。极限集的形状很好地描述了不同方向上极值观测值之间的关系。本文研究如何敏感的极限形状是在底层的重尾分布的变化。Copula与多变量极值拟合得很好。根据Galambos定理，只要边际极大值收敛，Copula的上端点存在方向导数是多元极值收敛的充分必要条件。最大稳定极限分布的copula不依赖于边际。因此，利润率似乎在多变量极端情况下起着辅助作用。本文提出的理论和例子对边距的重要性有不同的看法。对于轻尾Meta分布，其渐近行为对底层重尾原始分布的扰动非常敏感，即使重尾密度的渐近行为不受影响，它也可能发生剧烈变化。
英文摘要：
The paper focuses on a class of light-tailed multivariate probability distributions. These are obtained via a transformation of the margins from a heavy-tailed original distribution. This class was introduced in Balkema et al. (J. Multivariate Anal. 101 (2010) 1738-1754). As shown there, for the light-tailed meta distribution the sample clouds, properly scaled, converge onto a deterministic set. The shape of the limit set gives a good description of the relation between extreme observations in different directions. This paper investigates how sensitive the limit shape is to changes in the underlying heavy-tailed distribution. Copulas fit in well with multivariate extremes. By Galambos's theorem, existence of directional derivatives in the upper endpoint of the copula is necessary and sufficient for convergence of the multivariate extremes provided the marginal maxima converge. The copula of the max-stable limit distribution does not depend on the margins. So margins seem to play a subsidiary role in multivariate extremes. The theory and examples presented in this paper cast a different light on the significance of margins. For light-tailed meta distributions, the asymptotic behaviour is very sensitive to perturbations of the underlying heavy-tailed original distribution, it may change drastically even when the asymptotic behaviour of the heavy-tailed density is not affected.
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