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I have read that if the distribution of [itex]X[/itex] belongs to the exponential family, then the maxima drawn from a sample of [itex] X [/itex] follow:

[itex] f(x) = \frac{1}{\beta} exp{(-\frac{x-\mu}{\beta} - exp{(-\frac{x-\mu}{\beta})})} [/itex]

in the limit of infinetely large sample. The above is apparently called the Gumbel distribution.

First I wonder if I correctly imagine the limiting process. Let us say I construct a histrogramme in the following way (in fact I did it on my computer). I draw [itex] M [/itex] Gaussian random numbers. I determine their maximum [itex] m_1 [/itex]. Then I repeat the process [itex] N [/itex] times to get maxima [itex]m_1,\dots m_N [/itex]. I draw a histogramme of the [itex] m_i [/itex]. Must I imagine the limiting process as requiring both [itex] M,N \rightarrow \infty [/itex] ? Shouldn't increasing [itex]M[/itex] lead to larger and larger maximum values, thereby shifting the distribution of maxima to the right, to larger [itex]\mu[/itex]? How can I then obtain well defined [itex]\mu, \beta[/itex] in the limit?

Second, let us imagine the histogrammes for finite [itex] M,N [/itex] like I have actually drawn them. Let us characterize them by the [itex] \tilde\mu, \tilde \beta [/itex] of the Gumbel distribution which fits them best. I want to ask, how the parameters [itex]\tilde\mu, \tilde\beta[/itex] depend on [itex]M[/itex] and [itex]N[/itex]. Namely for the case where [itex] X [/itex] is a normal distribution of mean zero and variance 1.

In my example programme I can see my histogrammes move, but I could only try a few sample sizes. And I would like to understand the whole question logically. My histogrammes look very much like a Gumbel distribution, but I don't know what parameters to expect from theory and I want to be sure my programme does the right thing!

Any help will be much appreciated,

Angelos

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# Most probable extreme value depending on sample size

Can you offer guidance or do you also need help?

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