Generalized Entropy Index

From Open Risk Manual

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^{n}_{i=1} 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}{n a (a-1)} \sum_{i=1}^n \left( n^a w_i^a -n \right),& a \ne 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

Implementations

Open Source implementations of the Generalized Entropy index are available in

See Also

References