Gini representations of skewness, kurtosis and higher-order moments

Abstract

We provide pairwise-difference (Gini-type) representations of higher-order central moments for both general random variables and empirical moments. Such representations do not require a measure of location. For third and fourth moments, this yields pairwise-difference representations of skewness and kurtosis coefficients. We show that all central moments possess such representations, so no reference to the mean is needed for moments of any order. This is done by considering i.i.d. replications of the random variables considered, by observing that central moments can be interpreted as covariances between a random variable and powers of the same variable, and by giving recursions which link the pairwise-difference representation of any moment to lower order ones.

arXiv comment26 pages