A new index to measure the income inequality for the poor and the rich with application

Studying the characteristics of different income distributions, and compare the differences among different income distributions with respect to geographic regions, and changes over time periods is a very important area in economics. In order to compare the differences among different income distributions, different measures, including the Gini and Zenga indices have been developed. However, one of the limitations of the existing indices is that they only provide an overall measurement for the distribution, cannot provide the measure for any part of the distribution. However, comparing any different parts of the distribution or different distributions is very important. For example, we can compare the rich and the poor. Another limitation some existing measures including the Gini and Zenga indices is that they provide the same value for different distributions.To circumvent the above limitations, this paper proposes a new index which can make comparison between any groups with different income level, for example, the richest group and the poorest group. The new index provides different values for different distributions and different parts of any distributions. The new index satisfies the axioms of inequality measures. The statistical inferential results of the new index are derived and their performance is tested by a simulation study. We illustrate the applicability of our proposed index to analyze the income data of Hong Kong and compare different groups in the distribution