A Subgroups Decomposition of a New Inequality Index Proposed by Zenga

This paper concerns some notes about the inequality index, based on the ratios between lower and upper arithmetic means, recently proposed by Zenga [2007]. Given a non-negative variable observed on a population, we consider the case the population units can be partitioned in c subgroups. In this framework it is interesting to evaluate the relation between the overall uniformity (inequality) and the ones evaluated in each subgroup. In particular the common aim is usually to separate the within and between subgroups contributions to the overall uniformity (inequality). We initially deal with the decomposition of the overall uniformity in particular we propose, at first, a decomposition that allows to evaluate the contribution of each subgroup to the overall uniformity, this contribution comprises both the within and the between sources. We next introduce the evaluation of the point uniformity index either within the same subgroup or between two different subgroups such that a within/between subgroups uniformity decomposition arises. We show that the decomposition proposed leads to the representation of the overall uniformity index as a weighted average of two components related, respectively, to the within and to the between contributions. This last representation allows to obtain an analogous decomposition of the overall inequality index