I have a question regarding the Gumbel distribution. Based on the figure below, it appears that a form of the Fréchet distribution can be derived from it, using a log.
Variable $u_{i}$ represents an observed factor, and in contrast $z_{i}$ represents an unobserved factor that is i.i.d. And finally, $\mu$ represents the scale parameter of the Gumbel distribution.
Is it possible to provide the intuition or proof for obtaining this result?
