A New Metric for Quantifying the Tail Heaviness of Probability Distributions

Tail heaviness is an important characteristic of probability distributions. Effectively quantifying tail heaviness is crucial for constructing statistical models and comparing different distributions. Traditionally, kurtosis is employed as a metric for measuring tail heaviness; however, it suffers from several well-known limitations or drawbacks. Although several alternative measures have been proposed in the literature, how to appropriately quantify tail heaviness remains an unresolved issue. This paper proposes a novel method for measuring tail weight and introduces a new metric for quantifying tail heaviness, termed the “Tail Heaviness Index.” Our analysis of tail weight focuses primarily on the downside tail (i.e., the right tail) of unimodal distributions. First, we define the “Tail Probability Function” (TPF) as the normalized exceedance probability function. Subsequently, we define the tail weight as the product of the area under the TPF curve and the average density of the tail region. Furthermore, we define the tail heaviness index as the ratio of the tail weight of a given distribution to the tail weight of the benchmark exponential distribution. This paper presents closed-form expressions for the tail heaviness index for seven well-known distributions and compares them with the tail weight coefficient (TWC) proposed by Ortega[1].