Decomposition of gini and multivariate gini indices

A new type of decomposition by population subgroup is proposed for the Gini inequality index. The decomposition satisfies the completely identical distribution (CID) condition, whereby the between-group inequality is null if and only if the distribution within each subgroup is identical to all the others. Thus, this decomposition contrasts strikingly with the subgroup decomposition of the generalized entropy measures, which satisfy the condition that the between-group inequality is null if the mean within each subgroup equals those of all the others. The new decomposition can be generalized to the distance-Gini index and the volume-Gini index, two multivariate Gini indices introduced by Koshevoy and Mosler, with some modification of the index definition and a somewhat loosened CID condition in the latter case. The source decomposition is also generalized to these multi-dimensional indices. Interaction terms appear among sources of different attributes in the decomposition for the modified volume–Gini index.