Experiential learning: convergence of the GEV distribution to the Gumbel distribution

We learned here the mathematical formulation of the GEV distribution:

\(G(x) =
\begin{cases}
\exp \left\{ -\left[ 1 + \xi \left( \dfrac{x-\mu}{\sigma}\right)\right]^{-1/\xi}\right\},
& \text{if } 1 + \xi \left( \dfrac{x-\mu}{\sigma}\right) > 0,\ \xi \neq 0, \\
\exp \left\{ -\exp\left( -\dfrac{x-\mu}{\sigma}\right)\right\},
& \text{if } \xi = 0.
\end{cases} \)

We learned that the shape parameter \(\xi\) determines which extreme-value family a given GEV belongs. Gumbel is obtained for \(\xi \to 0\):

\( G(x) = \exp\left\{-\exp\left(-\dfrac{x-\mu}{\sigma}\right)\right\},
\quad x \in \mathbb{R}\)

Frechet is obtained for \(\xi > 0\):

\(G(x) = \exp\left\{-\left[1 + \xi \left(\dfrac{x-\mu}{\sigma}\right)\right]^{-1/\xi}\right\},
\quad x > \mu - \dfrac{\sigma}{\xi}\)

Weibull is obtained for \( \xi < 0\):

\( G(x) = \exp\left\{-\left[1 + \xi \left(\dfrac{x-\mu}{\sigma}\right)\right]^{-1/\xi}\right\},
\quad x < \mu - \dfrac{\sigma}{\xi}\)

However, it is not clear how the Gumbel distribution is obtained, for instance, from the Frechet distribution for \(\xi \to 0\), as the exponent \(-1/\xi\) tends to minus infinity and therefore the probability distribution degenerates. In fact, the R function "pfrechet" (or pFrechet", available in the R package DescTools) gives unreasonable estimates for \(\xi \to 0\). Try, for example:

and compare the estimated probabilities, by considering that in theory we should obtain the same values as the shape parameter of the Frechet is very close to 0. However, the results are actually very different, because the convergence of the Frechet (and the GEV as well) to Gumbel only holds after appropriate normalization (location and scale adjustment).

In detail, the Frechet distribution is given by (see above):

\(G(x) = \exp\left\{-\left[1 + \xi \left(\dfrac{x-\mu}{\sigma}\right)\right]^{-1/\xi}\right\},
\quad x > \mu - \dfrac{\sigma}{\xi}\)

Now rewrite: 

\(z = 1+\xi\frac{x-\mu}{\sigma}\),

which leads to:

\(G(z)=\exp\left(-z^{-1/\xi}\right)\).

We can also write:

\(z^{-1/\xi}=\exp^{-1/\xi\log z}\)

therefore obtaining 

\(G(z) = \exp\left(-e^{-1/\xi \log z}\right)\).

Now define a new variable: 

\(y = \frac{1}{\xi} \log z\)

Then:

\(G(y) = \exp(-e^{-y})\)

which is exactly the Gumbel probability distribution.

Let's verify the results by completing the above R code:

Last modified on March 24, 2026