3.5 Amato index (svyamato)

The Amato index is also based on the Lorenz curve, but instead of focusing on the area of the curve, it focuses on its length. Arnold (2012Arnold, Barry C. 2012. “On the Amato Inequality Index.” Statistics and Probability Letters 82 (8): 1504–6. http://EconPapers.repec.org/RePEc:eee:stapro:v:82:y:2012:i:8:p:1504-1506.) proposes a formula not directly based in the Lorenz curve, which 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.) uses to present the following estimator:

\[ \widehat{A} = \sum_{i \in S} w_i \bigg[ \frac{1}{\widehat{N}^2} + \frac{y_i^2}{\widehat{Y}^2} \bigg]^{\frac{1}{2}} \text{,} \]

which also generates the linearized formula for SE estimation.

The minimum value \(A\) assumes is \(\sqrt{2}\) and the maximum is \(2\). In order to get a measure in the interval \([0,1]\), the standardized Amato index \(\widetilde{A}\) can be defined as:

\[ \widetilde{A} = \frac{ A - \sqrt{2} }{2 - \sqrt{2} } \text{ .} \]

For additional usage examples of svyamato, type ?convey::svyamato in the R console.