3.6 Zenga Index and Curve (svyzenga, svyzengacurve)

The Zenga index and its curve were proposed in Zenga (2007Zenga, Michele. 2007. “Inequality Curve and Inequality Index Based on the Ratios Between Lower and Upper Arithmetic Means.” Statistica E Applicazioni 1 (4): 3–27.). As Polisicchio and Porro (2011Polisicchio, Marcella, and Francesco Porro. 2011. “A Comparison Between Lorenz L(P) Curve and Zenga I(P) Curve.” Statistica Applicata 21 (3-4): 289–301. http://hdl.handle.net/10281/49428.) noticed, this curve derives directly from the Lorenz curve, and can be defined as:

\[ Z(p) = 1 - \frac{L(p)}{p} \cdot \frac{1 - p}{1 - L(p)}. \]

In the convey library, an experimental estimator based on the Lorenz curve is used:

\[ \widehat{Z(p)} = \frac{ p \widehat{Y} - \widehat{\widetilde{Y}}(p) }{p \big[ \widehat{Y} - \widehat{\widetilde{Y}}(p) \big] }. \]

In turn, the Zenga index derives from this curve and is defined as:

\[ Z = \int_0^1 Z(p)dp. \]

However, its estimators were proposed by Langel (2012Langel, Matti. 2012. “Measuring Inequality in Finite Population Sampling.” PhD thesis. http://doc.rero.ch/record/29204.) and Barabesi, Diana, and Perri (2016Barabesi, Lucio, Giancarlo Diana, and Pier Francesco Perri. 2016. “Linearization of Inequality Indices in the Design-Based Framework.” Statistics 50 (5): 1161–72. doi:10.1080/02331888.2015.1135924.). In this library, the latter is used and is defined as:

\[ \widehat{Z} = 1 - \sum_{i \in S} w_i \bigg[ \frac{ ( \widehat{N} - \widehat{H}_{y_i} ) ( \widehat{Y} -\widehat{K}_{y_i} ) } { \widehat{N} \cdot \widehat{H}_{y_i} \cdot \widehat{K}_{y_i} } \bigg] \]

where \(\widehat{N}\) is the population total, \(\widehat{Y}\) is the total income, \(\widehat{H}_{y_i}\) is the sum of incomes below or equal to \(y_i\) and \(\widehat{N}_{y_i}\) is the sum of incomes greater or equal to \(y_i\).

For additional usage examples of svyzenga or svyzengacurve, type ?convey::svyzenga or ?convey::svyzengacurve in the R console.