Talk:Generalized entropy index

Transformation seems wrong

"The transformation is $A=1-e^{-GE}$ " seems wrong. Defining

$\phi ={\frac {1}{N}}\sum _{i=1}^{N}\left({\frac {y_{i}}{\mu }}\right)^{\alpha }$

I get (using $\varepsilon =1-\alpha$):

$A=1-\phi ^{1/\alpha }$

and

$GE={\frac {\phi -1}{\alpha (\alpha -1)}}$

which won't give the above relationship. (A is the Atkinson index) PAR (talk) 02:25, 25 May 2014 (UTC)

It is definitely correct for a GE index of order 0.

See here: https://www.academia.edu/1816869/A_note_on_the_relationship_between_the_Atkinson_index_and_the_generalised_entropy_class_of_decomposable_inequality_indexes_under_the_assumption_of_log-normality_of_income_distribution_or_volatility

The case of general $\epsilon$ also seems wrong, and disagrees with the definition given here: https://en.wikipedia.org/wiki/Atkinson_index

I believe it should be $A=1-{\sqrt[{\alpha }]{GE\varepsilon (\varepsilon -1)+1}}$.

Where does the 1/(alpha * (alpha - 1)) come from?

In the denominator of the fraction that is a factor of the constant by which the sum is multiplied, where does the...

alpha * (alpha - 1) come from? What's the explanation for why that's there? — Preceding unsigned comment added by 97.82.116.234 (talk) 03:47, 30 April 2022 (UTC)

I’m the one who asked that question. My guess is that the purpose of dividing by that expression is to make the formula undefined if alpha = 0 or 1.

…so that the formula won’t apply with those values of alpha.

…because Theil-L & Theil-T are the intended meanings for ge(0) & ge(1). 2600:6C55:7900:2B8:6DFC:EF4C:CC93:C9D5 (talk) 07:57, 11 June 2024 (UTC)

Focus on income inequality seems restrictive

This is a comment about the focus/tone of this article, rather than any element of the content per se. It seems that generalized entropies have far-ranging applications (information theory, income inequality, biological diversity measures) and this article implies that their primary application or the domain in which they make the most sense is in defining income inequality. Would be great if future edits made clear that income inequality is an *example* and a topic of study where theoretical and conceptual advances on this topic have been made, but that generalized entropies have a wider array of potential applications. Muniche (talk) 11:46, 14 July 2022 (UTC)

Explanation for the 3 formulas is omitted.

I’ll add it. 2600:6C55:7900:2B8:6DFC:EF4C:CC93:C9D5 (talk) 07:16, 11 June 2024 (UTC)

I said, in my addition to the article, that alpha represents an integer. Below in the article, it says that alpha can be any real number. I didn’t know that non-integer values are used. Maybe “integer” should be corrected to “real-number”.

I claim that the most meaningful & relevant ge for income-inequality is ge(-1).

I claim that that is the meaningful summation-aggregation inequality index.

Tomorrow at this talk-page, I I’ll tell why.

Later, unless there’s disagreement, I’d like to add that statement to the article. 2600:6C55:7900:2B8:6DFC:EF4C:CC93:C9D5 (talk) 07:49, 11 June 2024 (UTC)