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.

翻译：本文针对一般随机变量及经验矩，提出了高阶中心矩的成对差分（基尼型）表示方法。此类表示无需依赖位置度量。对于三阶与四阶矩，该框架可导出偏度与峰度系数的成对差分表示。我们证明所有中心矩均具备此类表示，这意味着任意阶矩的计算均无需参照均值。该结论通过以下方式实现：考虑随机变量的独立同分布重复抽样，观察到中心矩可解释为随机变量与其自身幂次之间的协方差，并给出将任意阶矩的成对差分表示与低阶矩相联系的递推关系。由此推导出数值求和恒等式。通过类似方法，我们建立了适用于一般随机变量的拉格朗日恒等式与比内-柯西恒等式的对应形式，并给出协方差经典柯西-施瓦茨不等式的简洁推导。最后，讨论了该方法在中心矩无偏估计中的应用。