How Much Data Do You Need? An Operational, Pre-Asymptotic Metric for Fat-tailedness

TL;DR

This paper introduces an operational measure of fat-tailedness for univariate distributions, focusing on finite-sample behavior to aid in practical statistical assessments and comparisons across distribution classes.

Contribution

It proposes a novel, heuristic metric based on the convergence rate of the Law of Large Numbers to compare distributions with finite first moments.

Findings

Provides explicit expressions and bounds for key distributions

Enables comparison of sums of different fat-tailed distributions

Addresses limitations of traditional tail index and kurtosis measures

Abstract

This note presents an operational measure of fat-tailedness for univariate probability distributions, in $[0,1]$ where 0 is maximally thin-tailed (Gaussian) and 1 is maximally fat-tailed. Among others,1) it helps assess the sample size needed to establish a comparative $n$ needed for statistical significance, 2) allows practical comparisons across classes of fat-tailed distributions, 3) helps understand some inconsistent attributes of the lognormal, pending on the parametrization of its scale parameter. The literature is rich for what concerns asymptotic behavior, but there is a large void for finite values of $n$, those needed for operational purposes. Conventional measures of fat-tailedness, namely 1) the tail index for the power law class, and 2) Kurtosis for finite moment distributions fail to apply to some distributions, and do not allow comparisons across classes and parametrization, that is between power laws outside the Levy-Stable basin, or power laws to distributions in other classes, or power laws for different number of summands. How can one compare a sum of 100 Student T distributed random variables with 3 degrees of freedom to one in a Levy-Stable or a Lognormal class? How can one compare a sum of 100 Student T with 3 degrees of freedom to a single Student T with 2 degrees of freedom? We propose an operational and heuristic measure that allow us to compare $n$-summed independent variables under all distributions with finite first moment. The method is based on the rate of convergence of the Law of Large numbers for finite sums, $n$-summands specifically. We get either explicit expressions or simulation results and bounds for the lognormal, exponential, Pareto, and the Student T distributions in their various calibrations --in addition to the general Pearson classes.