Angelos K
Dec29-11, 10:09 AM
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.:smile: 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
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.:smile: 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