A note on the kernel estimation of Atkinson’s index of inequality

World Bank macrodata of every country on our planet show that national incomes per capita account for a major part of population disparity and these follow usual distributions well known in the literature in almost all continents. Measuring and comparing disparity is a huge task that requires assembling the relative nature of small and large national incomes without distinctions. This is the main reason that pushes us to consider in this paper the Atkinson inequality index (continuous case) developed towards the end of the 20th century in order to measure this disparity. Since then its nonparametric estimator has not been developed but rather a well-known classical discrete form. This makes the estimation or measurement of economic inequalities not very complex. In this paper, we construct a kernel estimator of the Atkinson inequality index and by extension that of its associated welfare function, then we establish their almost sure asymptotic convergences. Finally, we explore the performance of our estimators in a simulation study and draw conclusions about national incomes per capita on each continent and then globally by making comparisons with the classical form, based on World Bank staff estimates based on sources and methods from ”The Changing Wealth of Nations”. The results obtained highlight the interest of kernel-based measures as well as the sensitivity of the index with respect to the inversion parameter.