Assume that we only consider $$G(x)=\exp(-\exp(\frac{x-\mu}{\sigma}))$$ is the Gumbel distribution.

***Question:*** Suppose we have a set of maximum values $\{Y_i\}_{i=1}^m$, why can the article directly (for example: [here][1]) use this set of maximum values to fit the Gumbel distribution based on the maximal likelihood method
?

By extreme value theorem, we only know that
$$
P(\frac{Y_n-b_n}{a_n}\le x)=G(x)
$$
but not
$
P(Y_n\le x)=G(x).
$

So I'm a bit confused about fitting the extreme value distribution directly with $Y_i$ for $i=1,\dots, m$, don't we need to normalize it?


  [1]: https://www.stat.purdue.edu/~huang251/1018.html