尾部从何处开始？以拐点和最大曲率点作为边界
Where does the tail start? Inflection Points and Maximum Curvature as Boundaries
AI总结：
针对单峰分布尾部起点缺乏界定的问题，本文提出用密度函数的拐点和最大曲率点划分主体与尾部，并讨论其估计及与峰度、极端分位数的比较。
摘要：
理解分布的尾部行为在统计理论中至关重要。例如，分布的尾部在极值统计中无处不在，并与极端事件发生的可能性相关。根据尾函数 $\bar{F}(x)=P(X>x)$ 在 $x\to\infty$ 时的行为，已有若干方法可以刻画分布的尾部。然而，对于单峰分布而言，分布的主体在何处结束、尾部又从何处开始？本文处理这一尚未解决的问题，并探索利用连续随机变量密度函数导数得到的分界点，即拐点和最大曲率点。这些点被用来划分分布的主体与其尾部。我们讨论了这些分界点的估计，并将其与峰度和极端分位数等其他和分布尾部相关的度量进行比较。我们推导了若干已知分布的所提出分界点，并表明该点可以成为定义分布尾部起点的合理准则。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
