Where Does the Tail Start? Inflection Points and Maximum Curvature as Boundaries

Abstract

Understanding the tail behaviour 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, (Formula presented.), behaves when (Formula presented.). However, for unimodal distributions, where does the core of the distribution end and where does 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 kurtosis and extreme quantiles. We derive the proposed delimiting points for several known distributions and show that it can be a reasonable criterion to define the starting point of the tail of a distribution.