## 3.7 J-Divergence and Decomposition (svyjdiv, svyjdivdec)

The J-divergence measure (Rohde 2016Rohde, Nicholas. 2016. “J-Divergence Measurements of Economic Inequality.” Journal of the Royal Statistical Society: Series A (Statistics in Society) 179 (3): 847–70. https://doi.org/10.1111/rssa.12153.) can be seen as the sum of $$GE_0$$ and $$GE_1$$, satisfying axioms that, individually, those two indices do not. Using $$U_\gamma$$ and $$T_\gamma$$ functions defined in Biewen and Jenkins (2003)Biewen, Martin, and Stephen Jenkins. 2003. “Estimation of Generalized Entropy and Atkinson Inequality Indices from Complex Survey Data.” Discussion Papers of DIW Berlin 345. DIW Berlin, German Institute for Economic Research. http://EconPapers.repec.org/RePEc:diw:diwwpp:dp345., the estimator can be defined as:

\begin{aligned} J &= \frac{1}{N} \sum_{i \in S} \bigg( \frac{ y_i - \mu }{ \mu } \bigg) \log \bigg( \frac{y_i}{\mu} \bigg) \\ \therefore \widehat{J} &= \frac{\widehat{T}_1}{\widehat{U}_1} - \frac{ \widehat{T}_0 }{ \widehat{U}_0 } \end{aligned}

Since it is a sum of two additive decomposable measures, $$J$$ itself is decomposable.

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