I know that the maximum of iid random variables has Gumbel distribution. For example, this is true when iid random variables have Exponential distribution.
Does this hold when iid random variables have Gumbel distribution themselves?
Let's say there are $N$ iid random variables with distribution $X_n\sim Gumbel(\mu, \beta),~n=1,…,N$. What is the distribution of $\text{max}_n~X_n$? If it is also a Gumbel distribution, say $Gumbel(\mu_{\text{max}}, \beta_{\text{max}})$, how are parameters $\mu_{\text{max}}$ and $\beta_{\text{max}}$ related to $\mu$ and $\beta$?
I tried to find an answer in multiple textbooks, but could not find anything.
Best Answer
Yes, due to the condition "iid".
The CDF of $X_i \sim {\rm Gumbel}(\mu,\beta)$ is $$ F_i(x) = \exp(-\exp((\mu - x)/\beta)).$$ Then the CDF for $Y_n = \max_{i = 1,\dots,n} X_i$ is $$\begin{aligned} F_Y(y) &= \prod_{i=1}^n P(X_i \le y) \\ &= \prod_{i=1}^n F_i(y) \\ &= \exp(-n\exp((\mu-y)/\beta)) \\ &= \exp(-\exp((\mu+\beta \ln n-y)/\beta)) \end{aligned}$$ Therefore, $Y_n \sim {\rm Gumbel}(\mu+\beta\ln n, \beta)$.