when searching for the distribution of the extreme value of a large set of iid Gaussian RVs, I stumbled upon this answer by Sasha, which gave more accurate results than the approximation given in Fisher & Tippett. There was no reference for the source of Sahsa's approximation, and it would be helpful if someone could direct me to some book or article that derives it.
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Sasha's answer cites
http://reference.wolfram.com/language/ref/ExtremeValueDistribution.html#6764486
The example there starting with the text "Construct an approximation for the distribution of maximum value in a normal sample of size $n$:" is the computation shown in her answer.
What more reference are you wanting?
Eric Towers
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The cited page doesn't give the derivation of the constants $\mu_n = \Phi^{-1}\left(1-\frac{1}{n} \right) \qquad \qquad \sigma_n = \Phi^{-1}\left(1-\frac{1}{n} \cdot \mathrm{e}^{-1}\right)- \Phi^{-1}\left(1-\frac{1}{n} \right)$ it simply states the final result and show that it works. – avishay antman Nov 27 '16 at 19:24
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These are ancient results. See Gumbel 1935, eqn (14'), et seq. – Eric Towers Nov 27 '16 at 22:11
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Of course. How can one not be familiar with such elementary paper, published in a french statistics journal in 1935? – avishay antman Nov 27 '16 at 23:58
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@avishayantman : Or, as one can find immediately from the linked Wikipedia page, the paper by Gumbel introducing th Gumbel distribution, the $\xi = 0$ case of the generalized extreme value distribution. The referenced page also introduces the $\Phi^{-1}(1 - \frac{1}{n})$ and the GEVD page, Sasha's $F_{EV}$. – Eric Towers Nov 28 '16 at 16:05