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<p>Under the design-based approach, we carry out the analysis of the Theil index decomposition by focusing on the joint inference for the within-group and the between-group components. First, we express the two population components as statistical functionals in terms of a discrete measure. Subsequently, we derive the corresponding influence functions and obtain their properties. On the basis of such findings, we introduce estimators for the Theil index components and their variance-covariance matrix, as well as a pivotal quantity for implementing confidence ellipses. By means of a Monte Carlo simulation study, we show the suitable performance of the component estimators and the variance-covariance matrix estimator and assess the adequate coverage of the confidence ellipse. Finally, we apply our methodology to the data collected during the 2021 Household Budget Survey in Italy. More precisely, considering energy consumption at the household level as the target variable and adopting the NUTS&#xa0;2 grouping, we estimate the components of the Theil index and the corresponding variability for such survey data.</p>

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Theil index decomposition: the influence function approach

  • 期刊论文

  • 发表日期：

    2026年6月25日

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  • Lucio Barabesi,
  • Federico Crescenzi,
  • Lorenzo Mori

摘要

Under the design-based approach, we carry out the analysis of the Theil index decomposition by focusing on the joint inference for the within-group and the between-group components. First, we express the two population components as statistical functionals in terms of a discrete measure. Subsequently, we derive the corresponding influence functions and obtain their properties. On the basis of such findings, we introduce estimators for the Theil index components and their variance-covariance matrix, as well as a pivotal quantity for implementing confidence ellipses. By means of a Monte Carlo simulation study, we show the suitable performance of the component estimators and the variance-covariance matrix estimator and assess the adequate coverage of the confidence ellipse. Finally, we apply our methodology to the data collected during the 2021 Household Budget Survey in Italy. More precisely, considering energy consumption at the household level as the target variable and adopting the NUTS 2 grouping, we estimate the components of the Theil index and the corresponding variability for such survey data.