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On Wikipedia (https://en.wikipedia.org/wiki/Pareto_distribution#Pareto_types_I–IV) one can find the relation between the different types of Pareto Distribution and the Generalized Pareto Distribution (Type IV). A more appropriate explanation on the relation between types of the distribution can be found in the book:

https://www.amazon.it/Distributions-Chapman-Monographs-Statistics-Probability-ebook/dp/B00UVB2PWY

The relationship is summarized as follows:

enter image description here

On the "Generalized Pareto Distribution" Wikipedia page (https://en.wikipedia.org/wiki/Generalized_Pareto_distribution) one can find a different formulation for the same distribution, in particular the survival function here is defined as:

$$ \overline{F}(x) = \left( 1 + \xi/\sigma \cdot ( x - \mu) \right)^{-1/\xi} $$

what is the connection between this survival function and:

$$ \overline{F}(x) = \left[ 1 + \left( 1/\sigma \cdot ( x - \mu) \right) ^{1/\gamma} \right]^{-\alpha} \text{?} $$

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  • $\begingroup$ The two families differ if and only if $\gamma\ne1$ $\endgroup$
    – Xi'an
    Commented Apr 16, 2022 at 13:48
  • $\begingroup$ There also is a $\xi$ in the first equation that is missing in the second even if $\gamma\neq 1$ $\endgroup$
    – Barbab
    Commented Apr 16, 2022 at 13:52
  • $\begingroup$ No because the $\xi$ can be incorporated inside the $\sigma$, so this is not a genuine difference. $\endgroup$
    – Xi'an
    Commented Apr 16, 2022 at 14:47
  • $\begingroup$ Ok so if I estimate the first, then I can also reasonably calculate the parameters of the second, assuming that $\gamma = 1$? $\endgroup$
    – Barbab
    Commented Apr 16, 2022 at 15:02
  • $\begingroup$ If you estimate $\alpha$ and $\sigma$ for the second model, the reparameterisation is $\xi=1/\alpha$ and $\sigma'=\alpha\sigma$ when $\gamma=1$. $\endgroup$
    – Xi'an
    Commented Apr 16, 2022 at 15:04

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