Maximum likelihood and Gumbel distribution. Does the likelihood have a global maximum?

Question

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.

Answer 1

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 $[\mu,\,\alpha]$.

Answer 2

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))