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Let $X_1,X_2,\dots,X_n$ be iid random variables with distribution function $F(x)$ and $M_n:=\max\{X_1,\dots,X_n\}$. By the extreme value theorem, there exist two sequences of real numbers $a_n>0$ and $b_n$ such that the following limits converge to a non-degenerate distribution function $G(x)$.

$$ G(x;\mu,\sigma)=P(X\le x; \mu. \sigma)=\exp(-\exp(\frac{x-\mu}{\sigma})) $$ for $x\in R$.

Q1: Why does Gumbel have two different expressions?


Also, the Gumbel, Frechet, and Weibull are from the family of extreme value distributions: $$ G_{\gamma}=\exp(-(1+\gamma x)^{-1/\gamma}) $$ for $1+\gamma x>0$.

Why this paper wrote in another version as follows $$ G(x;\mu, \sigma, k)=\exp(-[1-\frac{k(x-\mu)}{\sigma}]^{1/k}) $$

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    $\begingroup$ The "first" Gumbel is a particular case of the "second" Gumbel. $\endgroup$
    – Xi'an
    Oct 31, 2022 at 20:45

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Quoting from Wikipedia:

In probability theory and statistics, the Gumbel distribution (also known as the type-I generalized extreme value distribution) is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions.

It's not always clear whether the maximum or minimum version of the Gumbel is being used. One is obtained by reversing the sign of $x$ from the other. The version with $\mu$ and $\sigma$ allows for different location and scale parameters than the standard Gumbel.

For the generalized extreme value distribution, the definition of the shape parameter ($\gamma$ or $k$ in your examples) has two different conventions, one the negative of the other. See this page for an example of the resulting confusion.

As to "why" there are different parameterizations, I suppose it has to do with historical patterns of use of minimum versus maximum Gumbel distributions and their extensions in different fields of interest. For example, the standard parameterization of an accelerated-failure-time Weibull survival model is based on the minimum extreme value Gumbel, leading to errors if you try to use the maximum, which is sometimes the generic default.

Those differences are not going to go away. For all of the distributions you note, it's critical to be sure that you know precisely which parameterization is being used.

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  • $\begingroup$ Thanks, but I didn't get it. The difference between the two expressions is that one is normalized and the other is not. In practice, does it matter? $\endgroup$
    – Hermi
    Nov 14, 2022 at 19:47
  • $\begingroup$ @Hermi you have to be careful whether you are using the maximum- or minimum-extreme-value form of the asymmetric Gumbel distribution. See this page for the type of error that occurs if you mix them up, trying to get a Weibull distribution, derived from a minimum-extreme-value Gumbel, with the maximum-extreme-value form instead. So these differences can matter in practice. Always be careful to find out what form someone is using; be consistent in your own use. $\endgroup$
    – EdM
    Nov 14, 2022 at 20:34
  • $\begingroup$ What about the case where only the maximum value is considered here? Or will there be two expressions? One without $\mu, \sigma$, and one with these two parameters. $\endgroup$
    – Hermi
    Nov 14, 2022 at 20:37
  • $\begingroup$ @Hermi $\mu$ and $\sigma$ are location and scale parameters, respectively, that allow flexibility in fitting the general form of a distribution to data. The "standard" form of such a distribution has $\mu=0$ and $\sigma=1$. The form including $\mu$ and $\sigma$ is thus most general. If you know that you are using a "standard" form, as in getting a Weibull distribution from a minimum-extreme-value Gumbel, then it's simpler just to omit $\mu$ and $\sigma$ from the formula. $\endgroup$
    – EdM
    Nov 14, 2022 at 20:46

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