Generalized entropy index
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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. 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 (TT) 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.
The formula is
where is the income for individual i that is part of N and is the weight given to distances between incomes at different parts of the income distribution. Sometimes a different notation is used where = ß + 1.
For lower values of a close to 0, GE is more sensitive to changes in the lower incomes and vice versa for values closer to 1. Theil's TT is a = 1 and Theil's TL (also called mean log deviation) is a = 0. When a = 2 the value is half the square of the coefficient of variation:
The generalized entropy index is a transformation of the Atkinson index where . The transformation is A=1-e^(-GE), so that the Atkinson index is a probability instead of entropy.
When the of is replaced with (for example, income/person becomes persons/income) then is equivalent to .
- Theil index
- Atkinson index
- Lorenz curve
- Gini coefficient
- Hoover index
- Robin Hood index
- Suits index
- Income inequality metrics
- Rényi entropy
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