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I am interesting in determining if there is a closed form expression of the CDF and PDF of the maximum on $N$ Gumbel distributions that are independent but not identically distributed.

In particular, I am hoping to calculate $E(max(x_1,...,x_N))$

where $x_1,...,x_N$ are each random variables with $x_i \sim Gumbel(\mu_i,\beta_i)$.

When the variables are distributed iid gumbel, it turns out that the maximum also has a gumbel distribution and the expectation above has a closed form solution. I was wondering if any similar results exist for the inid (independent not identically distributed) case, but where each random variable follows a gumbel distribution?

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  • $\begingroup$ Do you have any special structure on the parameters $\mu_i, \beta_i$? $\endgroup$ Sep 21, 2015 at 19:50
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    $\begingroup$ Hi @kjetilbhalvorsen. Thanks for asking. I would prefer no structure, but it would still be very useful for me if we assumed $\mu_i = 0~ \forall ~ i$ $\endgroup$
    – johneric
    Sep 21, 2015 at 21:21
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    $\begingroup$ I doubt there's a nice closed form. The maximum CDF has distribution $\exp[-\sum_{i}\exp(-(x-\mu_i)/\beta_i)].$ One idea is to write the inner sum of exponentials as a hypergeometric function, and then look it up in an integral table. Try looking at Meijor G functions. $\endgroup$
    – Alex R.
    Sep 21, 2015 at 23:30

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The maximum CDF is: $$ \Phi(x)=\exp\bigg[-\sum_i \exp(-\frac{x-\mu_i}{\beta_i})\bigg] $$ which gives the maximum PDF: $$\phi(x) = \frac{\partial\Phi}{\partial x} = \bigg[\sum_i \frac {\exp\big(-\frac{x-\mu_i}{\beta_i}\big)}{\beta_i}\bigg]\Phi (x)$$

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