Inference by linearization for Zenga’s new inequality index: a comparison with the Gini index

Abstract

Zenga's new inequality curve and index are two recent tools for measuring inequality. Proposed in 2007, they should thus not be mistaken for anterior measures suggested by the same author. This paper focuses on the new measures only, which are hereafter referred to simply as the Zenga curve and Zenga index. The Zenga curve Z (α) involves the ratio of the mean income of the 100α% poorest to that of the 100(1 − α)% richest. The Zenga index can also be expressed by means of the Lorenz Curve and some of its properties make it an interesting alternative to the Gini index. Like most other inequality measures, inference on the Zenga index is not straightforward. Some research on its properties and on estimation has already been conducted but inference in the sampling framework is still needed. In this paper, we propose an estimator and variance estimator for the Zenga index when estimated from a complex sampling design. The proposed variance estimator is based on linearization techniques and more specifically on the direct approach presented by Demnati and Rao. The quality of the resulting estimators are evaluated in Monte Carlo simulation studies on real sets of income data. Finally, the advantages of the Zenga index relative to the Gini index are discussed.

Enlarging Regional Disparities in Energy Intensity within China

“…One of the most obvious advantages of the Zenga inequality index is that it can clearly show which part in the distribution contributes most to the overall disparity (Langel & Tillé, 2012; Pasquazzi & Zenga, 2018). We create disparity curves of energy intensity from 1995 to 2017 to show this property.…”

Components of Gini, Bonferroni, and Zenga Inequality Indexes for EU Income Data

“…Polisicchio (2008), Polisicchio and Porro (2009), Porro (2008), andPorro (2011) deal with properties of the curve defined by the point inequality measures I i and its relation with the Lorenz curve. Inferential problems related to the I index have been analyzed in Greselin and Pasquazzi (2009), Greselin et al (2010), Langel and Tille ´(2012), Antal et al (2011), andGreselin et al (2014). As for decomposition rules, Radaelli (2008a) proposed a subgroups decomposition for the point inequality indexes I i and the synthetic I index that has been applied to income data in Radaelli (2007), Radaelli (2008b), and and that has been compared with a subgroups decomposition rule for Gini's index in Radaelli (2010).…”

Linearization of inequality indices in the design-based framework

“…2012b). Langel and Tillé (2012b) have introduced the finite-population version of the Zenga new index based on smoothed quantiles and have provided the linearization of the corresponding functional. However, by considering the continuous-population expression as proposed by Zenga (2007, expression (5.6) in his paper), the "natural" finite-population counterpart of the Zenga new index may be given as the functional…”