The coefficient of variation, which measures the variability of a distribution from its mean, is not uniquely defined in the multidimensional case, and so is the multidimensional Gini index, which measures the inequality of a distribution in terms of the mean differences among its observations. In this paper, we connect these two notions of sparsity, and propose a multidimensional coefficient of variation based on a multidimensional Gini index. We demonstrate that the proposed coefficient possesses the properties of the univariate coefficient of variation. We also show its connection with the Voinov-Nikulin coefficient of variation, and compare it with the other multivariate coefficients available in the literature.

翻译：变异系数用于度量分布相对于其均值的变异性，但在多维情形下并无唯一定义；同样地，多维基尼指数——通过观测值间的平均差异来衡量分布的不平等性——也存在此问题。本文建立了这两种稀疏性度量概念间的联系，并提出一种基于多维基尼指数的多维变异系数。我们证明所提出的系数具备单变量变异系数的所有性质，同时阐明其与Voinov-Nikulin变异系数的关联，并与文献中现有的其他多元变异系数进行比较。