On the decomposition by subpopulations of the point and synthetic Zenga (2007) inequality indexes

The Radaelli (Stat Appl V I(2):117–136, 2008) decomposition by k subpopulations of the Zenga (Stat Appl V(1):3–27, 2007) point index is based on the decomposition of the point uniformity measure. In this work we decompose, in a more direct way, the point inequality in a weighted mean of $$k \times k$$ k × k relative differences between the upper mean of the subpopulation g and the corresponding lower mean of the subpopulation l; the weights are the product of their relative frequencies. From this decomposition, we obtain the decomposition of the point index into the within and the between components. The within component is given by the sum of k terms and the between component is the sum of $$k \times (k-1)$$ k × ( k - 1 ) 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 “relative frequencies” help in the interpretation of the $$3 \times 3$$ 3 × 3 contributions.