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I have a question regarding the extreme value distribution corresponding to i.i.d. samples $X_i$ from a normal distribution, say $X_i\sim N(\mu, \sigma^2)$.

According to the theorem of Fisher-Tippett-Gnedenko, the maximum follows a Gumbel distribution: $max(X_i)\sim Gumbel(\alpha, k)$ with location $\alpha$ and scale $k$. I would like to express these parameters in terms of $\mu$ and $\sigma$, i.e. as a function of the parameters of the underlying normal distribution.

I didn't find any clear indication in the literature how this can be done, and yet it should be simple. By playing around with simulations, where I simulated the $X_i$ for a certain range of $\sigma$ and $\mu$ and then estimate $\alpha$ and $k$ with the MLE, I found the following linear relationships:

$\alpha\approx 2.92\sigma+\mu$

$k\approx 0.306\sigma$

These relationships were estimated with the OLS and achieved a high $R^2$. They make intuitively sense to me, but I was wondering if and how they can be derived analytically.

My question is similar to (2) of this question from 2014, which was never answered though. Also, it bears some similarity to this one. However, in that In that article the question is to show convergence. My question is related to a different aspect, namely deriving the parameters of the distribution once we assume that convergence holds

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  • I don't know anything about extreme value theory, but as a starting point, you may want to show that $\left[\Phi\left(\dfrac{x-\mu}{\sigma}\right)\right]^n$ is the CDF of $\max_i(X_i)$, where $\Phi$ is the standard normal CDF. – Clarinetist Dec 18 '17 at 13:35

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