Linearization and Variance Estimation of the Bonferroni Inequality Index

Abstract

The study of income inequality is important for predicting the wealth of a country. There is an increasing number of publications where the authors call for the use of several indices simultaneously to better account for the wealth distribution. Due to the fact that income data are usuallyinequality measures, inference, influence function | 1009 DONG et al. | INTRODUCTIONNobel Prize-winning economist, Joseph Stiglitz, stated that income inequality is an important measure for forecasting the wealth of a country (Stiglitz, 2012). The most widely used inequality measure is the Gini index (Gini, 1912(Gini, ,1914. Since Corrado Gini suggested the index, it has been the subject of numerous publications. Its use is not restricted only to the economic field. It is surprising to note that even after a century, different applications of the Gini index pop up in new fields (see, e.g. Giorgi, 2019).Recent studies encourage the use of more than one inequality index simultaneously to better catch the inequality in different parts of the income distribution and thereby to better understand the socioeconomic reality and political significance of inequality (see, e.g. Osberg, 2017;Piketty, 2015). The most suitable candidate to place side by side with the Gini index could be the Bonferroni inequality index (Bonferroni, 1933). In fact, Pundir et al. (2005) show that both can be derived from the Lorenz Curve (Lorenz, 1905). Indeed, the two indices share several properties while maintaining some very interesting peculiarities.The opposition between the Bonferroni index and the Gini index is rooted when Carlo Emilio Bonferroni proposed his index in 1930. In the beginning, the Bonferroni index was fought by Corrado Gini and his followers who were very fond of the Gini index and who tried to avoid the use of any other measures that took the Gini index down the line (Giorgi, 1998). Only in the last 40 years, the Bonferroni index has been rediscovered by Piesch (1975) and Nygård and Sandström (1981). Several extensions and interpretations proposed for the Gini index (see, e.g. Giorgi, 2005, for a comprehensive review) have then been extended to the Bonferroni index, disclosing even more similarities and differences between the two indices.

Generalised Income Inequality Index

“…However, it has been re‐discovered 40 years later by Piesch (1975) and Nygård & Sandström (1981). The Bonferroni index shares a lot of properties with the Gini index, but it is more sensitive to the left tail of the income distribution than the Gini index (Pizzetti, 1951; Dong et al , 2021). Furthermore, several extensions and interpretations proposed for the Gini index hold also for the Bonferroni index (see Tarsitano, 1990, for a comprehensive review).…”

“…The linearised variables are then used in the expression of the variance estimator of the total estimator for estimating the sampling variance (see, e.g. Dong et al , 2021). The Graf method can be applied to almost all sampling designs as long as the expression of the variance estimator of the total estimator under the sampling design is known.…”