Theil index estimation by means of the influence function with an application to income surveys

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, on the basis of these findings, we introduce estimators of the Theil index and its variance. By means of a Monte Carlo study, we show that the variance estimator displays suitable performance in terms of bias and provides confidence intervals with adequate coverage. In addition, by considering such benchmarks, 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 conditions and inequalities of households and individuals after the Covid-19 pandemic.