On the decomposition of the Gini's mean difference and concentration ratio

In this paper we propose a decomposition for the Gini's mean difference of a real variate obtained as the sum of c variates observed on a finite population. The decomposition is mainly based on two different sortings of the values of each variate: one is the natural sorting of the values, the other one is obtained by ranking the values according to the corresponding totals. The obtained result is then applied to get a similar decomposition for the Gini's concentration ratio. This latter decomposition is then compared with other decompositions proposed in literature. Finally the decompositions for the Gini's mean difference and concentration ratio are extended to the more general case of a linear combination of variates.