Maximum likelihood and Gumbel distribution. Does the likelihood have a global maximum? It appears to me that if I move the mode $u$ more to the negative and increase the scale parameter $\alpha$, one can get always a higher likelihood. If this is true, is there a limit of the likelihood?
The Gumbel likelihood is given by
$$\log(L) = \sum_{i=1}^{N}\log \left[\dfrac{1}{\alpha}\exp(-y_i - e^{-y_i}) \right]$$
where $y_i = \dfrac{x_i-u}{\alpha}$
As User Xi'an points out, there appears to be a (local) maximum somewhere. However, I'm more asking about the global maximum and whether one exists.
 A: I will prove a more general result: if a density is log-concave, then
the log-likelihood of the corresponding location-scale family has a
global maximum. The wanted result then follows,
since the Gumbel density is log-concave.
Consider a univariate density $f^\star(y)$ which is log-concave and
smooth on the real line; the parameterized density using the location $\mu$ and the
scale $\alpha >0$ is
$$
   f(x;\,\mu,\,\alpha) := \frac{1}{\alpha} \, f^\star\left(\frac{x - \mu}{\alpha}\right).  
$$
We can use the following alternative parameter vector $[\nu,\,\beta]$
with $\beta >0$
$$
  \beta := 1 / \alpha, \qquad \nu := - \mu / \alpha.
$$
 We have a one-to-one smooth transformation $[\mu,\,\alpha] \mapsto [\nu,\,\beta]$. Using the parameter vector $[\nu,\,\beta]$, the density at $x$ writes as $\beta\,f^\star(\beta x + \nu)$ and the log-likelihood for a sample $X_i$ is
$$
    \log L = \sum_{i=1}^N \log\{ \beta \, f^\star(\beta X_i + \nu) \}.
$$
It is clear that this is a concave function of the vector
$[\nu,\,\beta]$ and hence that a global maximum exists (possibly for
infinite $\nu$ or $\beta$). But this implies that a global maximum
exists as well for the location-scale parameterization
$[\mu,\,\alpha]$.
A: Here is the (log-)likelihood surface in $(u,\alpha)$ for a sample of 100 points from a standard Gumbel:

As you can see, the mode is located near $(0,1)$ which is the true value of the parameter. The graph was made by the following R code
library(VGAM)
obs=rgumbel(1e3)
loca=seq(min(obs),max(obs),le=1e2)
scala=seq(.1*sd(obs),10*sd(obs),le=1e2)
like=matrix(0,1e2,1e2)
for (i in 1:1e2)
  for (j in 1:1e2)
   like[i,j]=sum(dgumbel(x=obs,loc=loca[i],scal=scala[j],log=TRUE))

