Pairwise Difference Representations of Moments: Gini and Generalized Lagrange identities

TL;DR

This paper introduces measure-free pairwise-difference representations for higher-order moments, including skewness and kurtosis, and extends classical identities like Lagrange and Binet-Cauchy to random variables, with applications to unbiased estimation.

Contribution

It provides novel measure-free pairwise-difference formulas for all central moments and extends classical identities to the probabilistic setting, enabling new unbiased estimation methods.

Findings

All central moments have pairwise-difference representations.

Derived analogues of classical identities for random variables.

Application to unbiased estimation of 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. Numerical summation identities are deduced. Through a similar approach, we give analogues of the Lagrange and Binet-Cauchy identities for general random variables, along with a simple derivation of the classic Cauchy-Schwarz inequality for covariances. Finally, an application to unbiased estimation of centered moments is discussed.