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.[citation needed] The primary use is for income inequality.[1] 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.

Formula[edit]

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 \alpha is the weight given to distances between incomes at different parts of the income distribution. Sometimes a different notation is used where \alpha = ß + 1.

For lower values of \alpha close to 0, GE is more sensitive to changes in the lower incomes and vice versa for values closer to 1. Theil's TT has \alpha=1 and the negative of the mean log deviation has \alpha=0. When \alpha=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}[verification needed (see talk page)], 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[edit]

References[edit]

  1. ^ Ullah, Aman ; Giles, David E. A. (1998) Handbook of Applied Economic Statistics, CRC Press. ISBN 0-8247-0129-1[page needed]