# Most probable extreme value depending on sample size

1. Dec 29, 2011

### Angelos K

Dear all,

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

$f(x) = \frac{1}{\beta} exp{(-\frac{x-\mu}{\beta} - exp{(-\frac{x-\mu}{\beta})})}$

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 $M$ Gaussian random numbers. I determine their maximum $m_1$. Then I repeat the process $N$ times to get maxima $m_1,\dots m_N$. I draw a histogramme of the $m_i$. Must I imagine the limiting process as requiring both $M,N \rightarrow \infty$ ? Shouldn't increasing $M$ lead to larger and larger maximum values, thereby shifting the distribution of maxima to the right, to larger $\mu$? How can I then obtain well defined $\mu, \beta$ in the limit?

Second, let us imagine the histogrammes for finite $M,N$ like I have actually drawn them. Let us characterize them by the $\tilde\mu, \tilde \beta$ of the Gumbel distribution which fits them best. I want to ask, how the parameters $\tilde\mu, \tilde\beta$ depend on $M$ and $N$. Namely for the case where $X$ 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

Last edited: Dec 29, 2011