尾巴从哪里开始?作为边界的拐点和最大曲率

Rafael Cabral, Maria de Iorio, Andrea Cremaschi
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引用次数: 0

摘要

了解分布的尾部行为在统计理论中至关重要。例如,分布的尾部在极端值统计中扮演着无处不在的角色,它与极端事件的可能性相关联。有几种方法可以根据尾函数 $\bar{F}(x) = P(X>x)$ 在 $x\to\infty$ 时的表现来描述分布尾部的特征。然而,对于单模态分布,分布的核心在哪里结束,尾部从哪里开始?本文解决了这一悬而未决的问题,并探讨了从连续随机变量密度函数的导数中得到的分界点(即拐点和最大曲率点)的用法。这些点用于划分分布的主体和尾部。我们讨论了这些限制点的估计,并将它们与其他与分布尾部相关的度量进行了比较,如峰度和极值量值。我们推导了几种已知分布的拟议临界点,并证明它是定义分布尾部起点的合理标准。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Where does the tail start? Inflection Points and Maximum Curvature as Boundaries
Understanding the tail behavior of distributions is crucial in statistical theory. For instance, the tail of a distribution plays a ubiquitous role in extreme value statistics, where it is associated with the likelihood of extreme events. There are several ways to characterize the tail of a distribution based on how the tail function, $\bar{F}(x) = P(X>x)$, behaves when $x\to\infty$. However, for unimodal distributions, where does the core of the distribution end and the tail begin? This paper addresses this unresolved question and explores the usage of delimiting points obtained from the derivatives of the density function of continuous random variables, namely, the inflection point and the point of maximum curvature. These points are used to delimit the bulk of the distribution from its tails. We discuss the estimation of these delimiting points and compare them with other measures associated with the tail of a distribution, such as the kurtosis and extreme quantiles. We derive the proposed delimiting points for several known distributions and show that it can be a reasonable criterion for defining the starting point of the tail of a distribution.
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