Jean-Marie Dufour, Abderrahim Taamouti, Meilin Tong
arXiv 26 Oct 2025 · Statistics — Methodology
arXiv:2510.22714 · PDF · DOI · OpenAlex · Extracted main text
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.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
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Showing the top 10 of 13 scored citations.