🤖 AI Summary
The empirical statistical behavior of sample skewness and kurtosis under small-sample regimes remains poorly understood.
Method: Leveraging asymptotic analysis, heavy-tailed distribution modeling, and experiments on both synthetic and real-world datasets, we derive rigorous theoretical bounds and examine moment estimation properties.
Contribution/Results: We establish, for the first time, a strict asymptotic lower bound on sample kurtosis as a function of sample size and skewness. Extending the Taylor power law to higher-order moments, we reveal that the skewness–kurtosis 4/3 scaling law arises intrinsically from the asymptotic structure of heavy-tailed distributions. This scaling is robust only for heavy-tailed data and sufficiently large samples (n ≳ 100); under small samples, kurtosis exhibits a pronounced negative bias with a nontrivial lower bound, rendering classical moment estimators severely biased. Our results provide a theoretically grounded criterion for the validity domain of moment-based estimation and correct the misconception of universal power-law applicability.
📝 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.