Generalized Entropy Index

Definition

For the purpose of measuring concentration, the Generalized Entropy Index is measure of concentration that draws from concepts of information theory

Details

If the total exposure is

$E_{T}=\sum _{i=1}^{n}E_{i}$

and the fractional exposures $w_{i}$ are defined as

$w_{i}={\frac {E_{i}}{E_{T}}}$

Then the Generalized Entropy index is defined as

$GE(\alpha )={\begin{cases}{\frac {1}{na(a-1)}}\sum _{i=1}^{n}\left(n^{a}w_{i}^{a}-n\right),&a\neq 0,1,\\\log(n)+\sum _{i=1}^{n}w_{i}\log(w_{i}),&\alpha =1,\\-\log(n)-{\frac {1}{n}}\sum _{i=1}^{n}\log(w_{i}),&\alpha =0.\end{cases}}$

Relation with the Theil Index

The Theil Index is the Generalized Entropy Index for a = 1

Usage

None

Variations

None

Issues and Challenges

None

Implementation

Open Source implementations of the Generalized Entropy index are available in

• the R package Ineq
• the Python library Concentration Library

See Also

• Shannon Index

References

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