Where does the tail start? Inflection Points and Maximum Curvature as Boundaries

TL;DR

This paper investigates how to precisely identify the boundary between the core and tail of unimodal distributions using inflection points and maximum curvature, providing a new method for tail delimitation.

Contribution

It introduces a novel approach to define the tail boundary of distributions based on derivatives of the density function, comparing it with existing tail measures.

Findings

Derived delimiting points for known distributions

Proposed criterion effectively identifies tail start

Compared with kurtosis and quantiles

Abstract

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.