The generalized entropy index has been proposed as a measure of income inequality in a population.[1] It is derived from information theory as a measure of redundancy in data. In information theory a measure of redundancy can be interpreted as non-randomness or data compression; thus this interpretation also applies to this index. In addition, interpretation of biodiversity as entropy has also been proposed leading to uses of generalized entropy to quantify biodiversity.[2]
The formula for general entropy for real values of
\alpha
(For the ge named “ge(alpha)”, where “alpha” represents an integer:
The 2nd formula below is for ge(1), also called “Theil-T”.
The 3rd formula below is for ge(0), also called Theil-L”.
The 1st formula below is for ge(alpha), for all integer alpha other than 0 & 1.)
where N is the number of cases (e.g., households or families),
yi
\alpha
\alpha
\alpha
An Atkinson index for any inequality aversion parameter can be derived from a generalized entropy index under the restriction that
\epsilon=1-\alpha
\alpha
The formula for deriving an Atkinson index with inequality aversion parameter
\epsilon
\epsilon=1-\alpha
Note that the generalized entropy index has several income inequality metrics as special cases. For example, GE(0) is the mean log deviation, GE(1) is the Theil index, and GE(2) is half the squared coefficient of variation.