A new interpretation of the Gini correlation

Abstract

We provide a new way of interpreting the Gini correlation in the income source decomposition of the Gini index as a linear function of the ratio of the concordance pseudo-Gini (which measures the degree of agreement between the ranking of the incomes within a particular source and the ranking with respect to total income) to the within-source Gini (which measures income disparities within a particular income source). As a by-product, we discuss the modifications of the stochastic approach to the Gini index to take survey weights into account. We also discuss the income source decomposition of the Gini index when the incomes are weighted.

Appendix: Proof that re-scaling the weights has no effect of the estimated Gini index

Consider the expression for the Gini index as given by Eq. (2) in this paper.

Let $\hat{G}(w)$ denote the overall Gini index when the weight assigned to $y_{i.}$ is $w_i$,$i=1,2,\ldots ,n$. Also let $\bar{{r}_{i.}} ^ {*} (w)=2\bar{r}_{i.} (w)-\left( {\sum \nolimits _{j=1}^n {w_j } }\right) -1$ be the transformed average rank vector when the weight assigned to $y_{i.}$ is $w_i$, $i=1,2,\ldots ,n$, where $\bar{r}_{1.} (w)=\left( {\frac{w_1 +1}{2}}\right)$; $\bar{r}_{2.} (w)=w_1 +\left( {\frac{w_2 +1}{2}}\right)$; $\bar{r}_{3.} (w)=w_1 +w_2 +\left( {\frac{w_3 +1}{2}}\right)$; $\bar{r}_{4.} (w)=w_1 +w_2 +w_3 +\left( {\frac{w_4 +1}{2}}\right)$; etc.

Let $\hat{G}(aw)$ denote the overall Gini index when the weight assigned to $y_{i.}$ is $\left( {aw_i }\right)$, $i=1,2,\ldots ,n$, where a is a suitable positive constant which is used to re-scale the weights. Also let $\bar{{r}_{i.}} ^ {*} (aw)=2\bar{r}_{i.} (aw)-\left( {\sum \nolimits _{j=1}^n {aw_j } }\right) -1$ be the transformed average rank vector when the weight assigned to $y_{i.}$ is $aw_i$, $i=1,2,\ldots ,n$, where $\bar{r}_{1.} (aw)=\left( {\frac{aw_1 +1}{2}}\right)$; $\bar{r}_{2.} (aw)=aw_1 +\left( {\frac{aw_2 +1}{2}}\right)$; $\bar{r}_{3.} (aw)=aw_1 +aw_2 +\left( {\frac{aw_3 +1}{2}}\right)$; $\bar{r}_{4.} (aw)=aw_1 +aw_2 +aw_3 +\left( {\frac{aw_4 +1}{2}}\right)$; etc.

It is easy to verify that $\bar{{r}_{i.}} ^ {*} (aw)=a\bar{{r}_{i.}} ^ {*} (w)$, $i=1,2,\ldots ,n$.

Hence, it follows from Eq. (2) that $\hat{G}(aw)=\left( {\frac{1}{\sum \nolimits _{j=1}^n {aw_j } }}\right) \frac{\sum \nolimits _{i=1}^n {\bar{{r}_{i.}} ^ {*} (aw)aw_i y_{i.} } }{\sum \nolimits _{i=1}^n {aw_i y_{i.} } }=\left( {\frac{1}{\sum \nolimits _{j=1}^n {aw_j } }}\right) \frac{\sum \nolimits _{i=1}^n {a\bar{{r}_{i.}} ^ {*} (w)aw_i y_{i.} } }{\sum \nolimits _{i=1}^n {aw_i y_{i.} } }=\left( {\frac{1}{\sum \nolimits _{j=1}^n {w_j } }}\right) \frac{\sum \nolimits _{i=1}^n {\bar{{r}_{i.}}^ {*} (w)w_i y_{i.} } }{\sum \nolimits _{i=1}^n {w_i y_{i.} } } =\hat{G}\left( w\right)$

This completes the proof.