Skewness-Kurtosis: small samples and power-law behavior

TL;DR

This paper investigates the empirical behavior of skewness and kurtosis in small samples and heavy-tailed distributions, establishing bounds and conditions for power-law scaling between these moments.

Contribution

It introduces a lower bound for sample kurtosis based on sample size and skewness, and extends Taylor power law to higher moments in the context of heavy-tailed data.

Findings

Power-law scaling between kurtosis and skewness occurs mainly in heavy-tailed distributions.

A lower bound for kurtosis as a function of sample size and skewness is established.

Scaling behavior is prominent in medium to large samples from heavy-tailed distributions.

Abstract

Skewness and kurtosis are fundamental statistical moments commonly used to quantify asymmetry and tail behavior in probability distributions. Despite their widespread application in statistical mechanics, condensed matter physics, and complex systems, important aspects of their empirical behavior remain unclear, particularly in small samples and in relation to their hypothesized power law scaling. In this work, we address both issues using a combination of empirical and synthetic data. First, we establish a lower bound for sample kurtosis as a function of sample size and skewness. Second, we examine the conditions under which the 4/3 power law relationship between kurtosis and skewness emerges, effectively extending Taylor power law to higher order moments. Our results show that this scaling behavior predominantly occurs in data sampled from heavy tailed distributions and medium, large sample sizes.