Most probable extreme value depending on sample size

[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

 

[mmwave]

Dear Angelos,

Thank you for sharing your thoughts and questions on the Gumbel distribution. I can see that you have put a lot of effort into understanding the concept and its application. Let me try to address your questions and provide some insights.

Firstly, you are correct in your understanding that the limiting process requires both M and N to approach infinity. This is because in order to accurately represent the distribution of maxima from a sample of X, we need a large enough sample size (N) and a large enough number of samples (M) to ensure that we are capturing a wide range of values from the distribution. As you mentioned, increasing M will shift the distribution of maxima to the right, but this is not a concern as long as we have a sufficiently large N to capture the true distribution.

In terms of obtaining well-defined values for \mu and \beta in the limit, this is where mathematical analysis and statistical techniques come into play. Through mathematical calculations and statistical methods, we can determine the parameters that best fit the Gumbel distribution to the data. This involves finding the values of \mu and \beta that minimize the difference between the observed data and the theoretical distribution.

Secondly, the parameters \tilde\mu and \tilde\beta will depend on both M and N. As M and N increase, the accuracy of the estimated parameters will also increase. This is because with more data points and more samples, we have a better representation of the true distribution, and thus, the parameters will be more precise.

I hope this helps to clarify your understanding of the Gumbel distribution and its application. It seems like you are on the right track with your program, and I encourage you to continue exploring and learning more about this concept. If you have any further questions, please feel free to reach out.