Lucio Barabesi
DEPS, USiena
Federico Crescenzi
Department of Economics, Engineering, Society and Business Administration, University of Tuscia
Lorenzo Mori
Department of Statistical Sciences, University of Bologna
Abstract
By assuming the design-based paradigm, an analysis of the Theil index and its estimation is carried out. First, by expressing the population Theil index as a statistical functional, we obtain its influence function and prove the corresponding properties. We also provide some new results on the influence function of the Gini index, which are suitable for a methodological comparison of the two inequality measures. Subsequently, based on these findings, we introduce estimators of the Theil index and its variance. A confidence band for the Theil index influence function is also proposed. By means of a simulation study, we show that the variance estimator displays suitable performance in terms of bias and provides confidence intervals with adequate coverage. In addition, the suggested variance estimation outperforms the corresponding methods based on the nonparametric and parametric bootstrap. An application of our achievements is considered by using the data from the “Survey on Vulnerability to Poverty” held in 2021 in Tuscany (Italy) with the goal to map the socio-economic inequalities of households after the Covid-19 pandemic distinguishing between production areas and provinces. The results highlight an increasing inequality moving from touristic areas to the ones where production is based on industries
Keywords
Design-based approach, Inequality measure, Influence function, Variance estimation
Jel Codes
C13; C15; E21
