Generalized entropy index

The generalized entropy index is a general formula for measuring redundancy in data. The redundancy can be viewed as inequality, lack of diversity, non-randomness, compressibility, or segregation in the data.^{[citation needed]} The primary use is for income inequality. It is equal to the definition of redundancy in information theory that is based on Shannon entropy when α = 1 which is also called the Theil index (T_T) in income inequality research. Completely "diverse" data has no redundancy so that GE=0, so that it increases in the opposite direction of a Diversity index. It increases with order rather than disorder, so it is a negated measure of entropy.

Formula

The formula is

$GE(\alpha) = \frac{1}{N \alpha ( \alpha - 1 )} \sum_{i=1}^N\left[ \left(\frac{y_i}{\overline{y}} \right)^\alpha - 1 \right], \quad \text{ for real values } \alpha \ne 0,1$

$GE(\alpha) = \frac{1}{N} \sum_{i=1}^N\left[ \frac{y_i}{\overline{y}} \ln\left(\frac{y_i}{\overline{y}} \right)\right], \quad\quad \text{ for } \alpha =1$

$GE(\alpha) = \frac{1}{N} \sum_{i=1}^N \ln\left(\frac{\overline{y}}{y_i} \right), \quad \quad\quad \text{ for } \alpha = 0,$

where $y_i$ is the income for individual i that is part of N and α is the weight given to distances between incomes at different parts of the income distribution. Sometimes a different notation is used where α  = β + 1.

For lower values of α close to 0, GE is more sensitive to changes in the lower incomes and vice versa for values closer to 1. Theil's T_T is α  = 1 and Theil's T_L (also called mean log deviation) is α  = 0. When α  = 2 the value is half the square of the coefficient of variation:

$GE(\alpha) = 1/2 (\sigma/\mu)^2 \quad \quad\quad \text{ for } \alpha = 2.$

The generalized entropy index is a transformation of the Atkinson index where $\epsilon=1-\alpha$. The transformation is A=1-e^(-GE), so that the Atkinson index is a probability instead of entropy.

When the $y_i$ of $\alpha=1$ is replaced with $1/y_i$ (for example, income/person becomes persons/income) then $\alpha=1$ is equivalent to $\alpha=0$.

See also

• Theil index
• Atkinson index
• Lorenz curve
• Gini coefficient
• Hoover index
• Robin Hood index
• Suits index
• Income inequality metrics
• Rényi entropy