On the decomposition by sub populations of the point ih(y ) and synthetic i(y ) inequality indexes

The Radaelli (2008) decomposition by k subpopulations of the Zenga (2007) point index is based on the decomposition of the point uniformity measure. In this work, we first obtain, by the use of the relation between the mean value M of a mixture and the means of the k subpopulations of the mixture, a k by k additive decomposition of the difference between the upper mean and the lower mean. Then, dividing both sides of the above mentioned decomposition we obtain a k by k additive decomposition of the point measure. From this latter decomposition, with simple "aggregations", we obtain a k additive decompositions of the point measure, and the decomposition of the point measure into the within and the between components. The decompositions proposed in this paper are applied to the net disposable income of the 8151 Italian households partitioned in three macroregions, supplied by the 2012 Bank of Italy sample survey on household income and wealth. This application shows that the values of the conditional relative frequencies a(g|h)and p(l|h) help in the interpretation of the 3 by 3 contributions B{hlg} as well as in the inter pretation of the 3 contributions B{hl.}.