Bounded Distributions place Limits on Skewness and Larger Moments

TL;DR

This paper establishes bounds on skewness and kurtosis for positive and bounded distributions, revealing fundamental limits on their statistical shape characteristics.

Contribution

It introduces bounds on skewness based on coefficient of variation and extends these bounds to kurtosis and potentially higher moments for bounded distributions.

Findings

Skewness $D_3$ is bounded below by $\, ext{CoV} - 1/ ext{CoV}$.

Bounds are extended to distributions with bounded support.

Conjectures bounds exist for higher moments.

Abstract

Distributions of strictly positive numbers are common and can be characterized by standard statistical measures such as mean, standard deviation, and skewness. We demonstrate that for these distributions the skewness $D_3$ is bounded from below by a function of the coefficient of variation (CoV) $\delta$ as $D_3 \ge \delta-1/\delta$. The results are extended to any distribution that is bounded with minimum value $x_{\rm min}$ and/or bounded with maximum value $x_{\rm max}$. We build on the results to provide bounds for kurtosis $D_4$, and conjecture analogous bounds exists for higher statistical moments.