Parametric and Parameter‐Free Shape Moments

Abstract

Skewness and kurtosis are commonly considered sufficient representatives of distributional shape, generally represented by the third and fourth standardized central moments (moments about the mean). Certain distributions own parametric shape moments, while others do not. In particular, the normal and exponential have parameter‐free shape moments. In this article, we offer an in‐depth explanation for this phenomenon, based on Pearson skewness measure and on a new “Random‐identity Paradigm,” recently published.