Measuring Income Inequality via Percentile Relativities

``The rich are getting richer'' implies that the population income distributions are getting more right skewed and heavily tailed. For such distributions, the mean is not the best measure of the center, but the classical indices of income inequality, including the celebrated Gini index, are all mean-based. In view of this, Professor Gastwirth sounded an alarm back in 2014 by suggesting to incorporate the median into the definition of the Gini index, although noted a few shortcomings of his proposed index. In the present paper we make a further step in the modification of classical indices and, to acknowledge the possibility of differing viewpoints, arrive at three median-based indices of inequality. They avoid the shortcomings of the previous indices and can be used even when populations are ultra heavily tailed, that is, when their first moments are infinite. The new indices are illustrated both analytically and numerically using parametric families of income distributions, and further illustrated using capital incomes coming from 2001 and 2018 surveys of fifteen European countries. We also discuss the performance of the indices from the perspective of income transfers