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：我们为一般随机变量及经验矩提供了高阶中心矩的成对差分（基尼型）表示。此类表示无需位置度量。对于三阶和四阶矩，这产生了偏度和峰度系数的成对差分表示。我们证明所有中心矩均具有此类表示，因此任何阶矩的计算均无需参考均值。这一结论通过以下方式实现：考虑所研究随机变量的独立同分布复制，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，并给出将任意矩的成对差分表示与低阶矩相联系的递归关系。数值求和恒等式得以推导。通过类似方法，我们给出了适用于一般随机变量的拉格朗日恒等式与Binet-Cauchy恒等式的类比形式，并提供了协方差经典柯西-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。