Meta densities and the shape of their sample clouds
Meta densities and the shape of their sample clouds
复制标题
元密度及其样本云的形状
DOI：
10.1016/j.jmva.2010.02.010
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发表时间：
2010
期刊：
影响因子：
--
通讯作者：
Natalia Lysenko
中科院分区：
文献类型：
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作者：
Guus Balkema;P. Embrechts;Natalia Lysenko
本文比较了两种多元密度的水平集的形状。密度是正的、连续的，并且具有相同的依赖结构。密度 f 是重尾的。它沿所有光线以相同的速率减小，直至达到正常数。 c↓0 的水平集 {f>c} 具有极限形状，即有界凸集。我们变换每个坐标以获得具有高斯边缘的新密度 g。我们还将考虑拉普拉斯密度 g 或对称威布尔边际密度。将证明新的轻尾密度 g 的水平集也具有极限形状，即有界星形集。该集合的边界可以明确地写为取决于两个正参数的简单方程的解。极限形状在极端值研究和风险理论中很有趣，因为它决定了不同方向的极端观测值如何相互关联。尽管密度 f 和 g 在构造上具有相同的系词，但水平集的形状并不相关。了解一种密度的水平集的极限形状并不能提供有关另一种密度的极限形状的信息。
This paper compares the shape of the level sets for two multivariate densities. The densities are positive and continuous, and have the same dependence structure. The density f is heavy-tailed. It decreases at the same rate–up to a positive constant–along all rays. The level sets {f>c} for c↓0, have a limit shape, a bounded convex set. We transform each of the coordinates to obtain a new density g with Gaussian marginals. We shall also consider densities g with Laplace, or symmetric Weibull marginal densities. It will be shown that the level sets of the new light-tailed density g also have a limit shape, a bounded star-shaped set. The boundary of this set may be written down explicitly as the solution of a simple equation depending on two positive parameters. The limit shape is of interest in the study of extremes and in risk theory, since it determines how the extreme observations in different directions relate. Although the densities f and g have the same copula–by construction–the shapes of the level sets are not related. Knowledge of the limit shape of the level sets for one density gives no information about the limit shape for the other density.