The generalized entropy index has been proposed as a measure of income inequality in a population.[1] It is derived from information theory as a measure of redundancy in data. In information theory a measure of redundancy can be interpreted as non-randomness or data compression; thus this interpretation also applies to this index. In addition, interpretation of biodiversity as entropy has also been proposed leading to uses of generalized entropy to quantify biodiversity.[2]
Formula
The formula for general entropy for real values of <math>\alpha</math> is:
<math display="block">GE(\alpha) = \begin{cases}
\frac{1}{N \alpha (\alpha-1)}\sum_{i=1}^N\left[\left(\frac{y_i}{\overline{y}}\right)^\alpha - 1\right],& \alpha \ne 0, 1,\\
\frac{1}{N}\sum_{i=1}^N\frac{y_{i}}{\overline{y}}\ln\frac{y_{i}}{\overline{y}},& \alpha=1,\\
-\frac{1}{N}\sum_{i=1}^N\ln\frac{y_{i}}{\overline{y}},& \alpha=0.
\end{cases}</math>
where N is the number of cases (e.g., households or families), <math>y_i</math> is the income for case i and <math>\alpha</math> is a parameter which regulates the weight given to distances between incomes at different parts of the income distribution. For large <math>\alpha</math> the index is especially sensitive to the existence of large incomes, whereas for small <math>\alpha</math> the index is especially sensitive to the existence of small incomes.
An Atkinson index for any inequality aversion parameter can be derived from a generalized entropy index under the restriction that <math>\epsilon=1-\alpha</math> - i.e. an Atkinson index with high inequality aversion is derived from a GE index with small <math>\alpha</math>. Moreover, it is the unique class of inequality measures that is a monotone transformation of the Atkinson index and which is additive decomposable. Many popular indices, including Gini index, do not satisfy additive decomposability.[1][3]
The formula for deriving an Atkinson index with inequality aversion parameter <math>\epsilon</math> under the restriction <math>\epsilon = 1-\alpha</math> is given by:
<math display="block">A=1-[\epsilon(\epsilon-1)GE + 1]^{(1/(1-\epsilon))} \qquad \epsilon\ne1</math>
<math display="block">A= 1-e^{-GE} \qquad \epsilon=1</math>
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