## 3.9 Rényi Divergence (svyrenyi)

Another measure used in areas like ecology, statistics and information theory is the Rényi divergence measure. Using the formula defined in Langel (2012Langel, Matti. 2012. “Measuring Inequality in Finite Population Sampling.” PhD thesis. http://doc.rero.ch/record/29204.), the estimator can be defined as:

$\widehat{R}_\alpha = \begin{cases} \frac{1}{\alpha - 1} \log \bigg[ \widehat{N}^{\alpha - 1} \sum_{i \in S} w_i \cdot \bigg( \frac{y_k}{ \widehat{Y} } \bigg) \bigg], &\text{if } \alpha \neq 1, \\ \sum_{i \in S} \frac{w_i y_i}{ \widehat{Y}} \log \frac{\widehat{N} y_i}{\widehat{Y}}, &\text{if } \alpha = 1, \end{cases}$

where $$\alpha$$ is a parameter with a similar economic interpretation to that of the $$GE_\alpha$$ index.

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