The Gini's mean difference was defined as the expected absolute difference between a random variable and its independent copy. The corresponding normalized version, namely Gini's index, denotes two times the area between the egalitarian line and the Lorenz curve. Both are dispersion indices because they quantify how far a random variable and its independent copy are. Aiming to measure dispersion in the multivariate case, we define and study new Gini's indices. For the bivariate case we provide several results and we point out that they are "dependence-dispersion" indices. Covariance representations are exhibited, with an interpretation also in terms of conditional distributions. Further results, bounds and illustrative examples are discussed too. Multivariate extensions are defined, aiming to apply both indices in more general settings. Then, we define efficiency Gini's indices for any semi-coherent system and we discuss about their interpretation. Empirical versions are considered in order as well to apply multivariate Gini's indices to data.

翻译：基尼平均差定义为随机变量与其独立副本之间的期望绝对差。相应的归一化版本，即基尼指数，表示平等线洛伦兹曲线之间面积的两倍。两者均为离散指数，因为它们衡量随机变量与其独立副本之间的偏离程度。为度量多元情况下的离散性，我们定义并研究了新型基尼指数。针对二元情形，我们给出了若干结果，并指出它们是“依赖-离散”指数。我们展示了协方差表示形式，并阐释了其在条件分布中的意义。进一步讨论了边界条件、说明性示例及其他相关结果。为在更一般情境下应用这两种指数，我们定义了多元扩展形式。随后，针对任意半协同系统定义了效率基尼指数，并探讨其解释意义。最后，为将多元基尼指数应用于数据，我们考虑了其经验版本。