Gini indices and the moments of the share density function

The expected value of the share density of the income distribution can be expressed in terms of the Gini index. The variance of the share density of the income distribution is interesting because it gives a relationship between the first and the second order Gini indices. We find an expression for this variance and, as a result, we obtain some nontrivial bounds on these Gini indices. We propose new statistics on the income distribution based on the higher moments of the share density function. These new statistics are easily computable from the higher order Gini indices. Relating these moments to higher order Ginis suggests new estimates on these quantities.