Generalized entropy

The generalized or generalized entropy or the generalized / generalized entropy index (abbreviation: GE ) is a general formula for measuring redundancy (of data). Redundancy can be viewed as disparity (inequality), lack of diversity, non-randomness , compressibility, or segregation of the data. This measure is mainly used as a measure of unequal distribution . It is similar to the definition of redundancy, which is based on Shannon entropy , if , in the unequal distribution measurement, it is also referred to as the Theil index . Completely different data have no redundancy, so that it follows that it is in the opposite direction of a disparity measure. This increases with order rather than disorder, so it is a negated measure of entropy. ${\displaystyle \alpha =1}$ ${\displaystyle T_{T}}$ ${\displaystyle GE=0}$

formula

The formula is:

${\displaystyle GE(\alpha )={\begin{cases}{\frac {1}{N\alpha (\alpha -1)}}\sum _{i=1}^{N}\left[\left({\frac {y_{i}}{\overline {y}}}\right)^{\alpha }-1\right]&\mathrm {f{\ddot {u}}r} {\text{ reelle Werte }}\alpha \neq 0,1\,,\\{\frac {1}{N}}\sum _{i=1}^{N}\left[{\frac {y_{i}}{\overline {y}}}\ln \left({\frac {y_{i}}{\overline {y}}}\right)\right]&\mathrm {f{\ddot {u}}r} \,\alpha =1\,,\\{\frac {1}{N}}\sum _{i=1}^{N}\ln \left({\frac {\overline {y}}{y_{i}}}\right)&\mathrm {f{\ddot {u}}r} \,\alpha =0\,,\end{cases}}}$

where the income of each individual who denotes a part of and is the weighting of the gaps between incomes at different parts of the income distribution represents. Sometimes a different notation is used. ${\displaystyle y_{i}}$ ${\displaystyle i}$ ${\displaystyle N}$ ${\displaystyle \alpha }$ ${\displaystyle \alpha =\beta +1}$

For lower values ​​of close to 0, the GE is sensitive for lower incomes and vice versa (vice versa) for values ​​of almost 1. This is at and the Theil-L index , the mean logarithmic deviation, is at . If is, the value is half the squared coefficient of variation : ${\displaystyle \alpha }$ ${\displaystyle T_{T}}$ ${\displaystyle \alpha =1}$ ${\displaystyle T_{L}}$ ${\displaystyle \alpha =0}$ ${\displaystyle \alpha =2}$

${\displaystyle GE(\alpha )=1/2(\sigma /\mu )^{2}\quad \quad \quad \mathrm {f{\ddot {u}}r} \,\alpha =2\,.}$

The GE is a transformation of the Atkinson measure , where applies. This transformation is such that the Atkinson measure is a probability rather than an entropy. ${\displaystyle \epsilon =1-\alpha }$ ${\displaystyle A=1-\exp(-GE)}$

If by by : (for example, income per person is changing to person depending income) replaced, then with equivalent. ${\displaystyle y_{i}}$ ${\displaystyle \alpha =1}$ ${\displaystyle {\tfrac {1}{y_{i}}}}$ ${\displaystyle \alpha =1}$ ${\displaystyle \alpha =0}$

See also

• Rényi entropy
• Lorenz curve
• Gini coefficient
• Hoover inequality
• Robin Hood Index
• Suits index

Individual evidence

1. ↑ Aman Ullah, David Evan Albert Giles: Handbook of Applied Economic Statistics. CRC Press, 1998. ISBN 0824701291 .