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For me, the usual definition of the Herfindahl index is this one. See also this question.

Here, on page 26, it is defined as a function of the coefficient of variation:

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Are these definitions of the Herfindahl index related to each other? Does anyone know an alternative source for the latter definition?

Thank you.

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    $\begingroup$ It helps to do a little algebra first, because you will see that $HI$ isn't really defined in terms of the CV: it's essentially a standardized version of the log variance of the $(R_i).$ The persistent logarithms make it clear it cannot be mathematically the same as the definitions you link to (which are rational functions of the $R_i,$ not logarithmic), highlighting the appropriateness of seeking some kind of relationship rather than an algebraic equivalence. $\endgroup$
    – whuber ♦
    Commented Aug 25, 2022 at 14:18
  • $\begingroup$ Thank you for your comment. This possible relationship is what I am interested in. I suppose the author had something in mind when naming this index the same as the usual Herfindahl index. $\endgroup$
    – Quodly
    Commented Nov 18, 2022 at 7:32

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