The paper suggests a new generalized variance concept for measuring multidimensional inequality of a stratified society, based on multivariate statistical methods, where the members of society form a cloud in the oblique space of dimensions of inequality, such as income, expenditure and property. The cloud presents the multidimensional inequality capsulized in the cloud. The goal is to condense all the inequality information embodied by the cloud into a composite compact metric characterizing both the shape and the inner structure of the cloud. Contrary to the conventional literature that considers multidimensionality as a unidimensional weighted combination of the dimensions, our new composite index measures the inequality of the configuration of the points in the cloud. Our aim is twofold. First, we introduce the Inequality Covariance Matrix (ICM) assigned to the cloud, with elements measuring the correlations among dimensions. Having ICM, we propose the Generalized Variance (GV) of ICM to measure the composite Generalized Variance Inequality (GVI) level. Second, to evaluate the stratum-specific structure of the overall inequality, we suggest a new two-stage procedure. In the first stage, we divide the total GVI into between-groups and within-groups effects. Then, in the second stage the contributions of the strata to the within-groups inequality and, the contributions of the dimensions to the between-groups inequality are calculated. This GVI approach is sensitive to the correlation system, decomposable into stratum effects and, the number of dimensions is not limited. Moreover, including the log-dimensions in the analysis, GVI yields an Entropy Covariance Matrix giving a new Generalized Variance Entropy index. Finally, the GVI of censored poverty indicators means multidimensional poverty measurement. This special complex task is not yet solved in the traditional literature so far.

本文基于多元统计方法提出了一种新的广义方差概念，用于衡量分层社会的多维不平等，其中社会成员在收入、支出和财产等不平等维度的倾斜空间中形成云。云呈现了云中封装的多维不平等。目标是将云所包含的所有不等式信息浓缩为一个复合紧凑度量，同时表征云的形状和内部结构。与将多维性视为维度的单维加权组合的传统文献相反，我们的新综合指数衡量云中点配置的不平等。我们的目标是双重的。第一的，我们引入了分配给云的不等式协方差矩阵 (ICM)，其中元素测量维度之间的相关性。有了 ICM，我们提出了 ICM 的广义方差 (GV) 来衡量复合广义方差不等式 (GVI) 水平。其次，为了评估整体不平等的特定阶层结构，我们建议采用新的两阶段程序。在第一阶段，我们将总 GVI 分为组间效应和组内效应。然后，在第二阶段，计算阶层对组内不平等的贡献，以及维度对组间不平等的贡献。这种 GVI 方法对相关系统敏感，可分解为层效应，并且维数不受限制。此外，包括分析中的对数维度，GVI 产生一个熵协方差矩阵，给出一个新的广义方差熵指数。最后，删减贫困指标的 GVI 意味着多维贫困测量。迄今为止，传统文献中还没有解决这个特殊的复杂任务。