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Proof or relationship between generalised entropy and Atkinson indexes

Hi, see attached this difficult proof in pdf format.

This result can be shown to be correct computationally, but is there an analytic solution?

Any guidance would be most appreciated.

Edit: see also these additional identities that may be useful.

Re: Proof or relationship between generalised entropy and Atkinson indexes

Hey frustrated.

Have you looked at the Central Limit Theorem? This theorem is basically what makes a lot of the statistical results work and it's also the basis for a lot of the Weiner Process and Brownian motion stuff in Finance and applied statistics that use this.

Re: Proof or relationship between generalised entropy and Atkinson indexes

Not really, but I will now in depth.

Edit: Ok CLT is now familiar, but have not looked at the proofs. We are already implicitly using this as there is an assumption of a smooth log-normal distribution.

Edit: it should be noted that this problem is scale invariant, so the real value of ybar is irrelevant.