Kernel estimation of the Quintile Share Ratio index of inequality for heavy-tailed income distributions

Evidence from micro-data shows that capital incomes are exceedingly volatile, which makes up a disproportionately high contribution to the overall inequality in populations with the heavy-tailed nature on the income distributions for many countries. The quintile share ratio (QSR) is a recently introduced measure of income inequality, also forming part of the European Laeken indicators and which cover four important dimensions of social inclusion (health, education, employment and financial poverty). In 2001, the European Council decided that income inequality in the European Union member states should be described using a number of indicators including the QSR. Non-parametric estimation has been developed on the QSR index for heavy-tailed capital incomes distributions. However, this method of estimation does not give satisfactory statistical performances, since it suffers badly from under coverage, and so we cannot rely on the non -parametric estimator. Hence, we need another estimator in the case of heavy-tailed populations. This is the reason why we introduce, in this paper, a class of semi-parametric estimators of the QSR index of economic inequality for heavy-tailed income distributions. Our methodology is based on the extreme value theory, which offers adequate statistical results for such distributions. We establish their asymptotic distribution, and through a simulation study, we illustrate their behavior in terms of the absolute bias and the median squared error. The simulation results clearly show that our estimators work well.