In this paper, the authors compare different methods of fitting generalized extreme value distribution (such as the maximum likelihood method). For example, the Gumbel distribution:
$$ F(x)=\exp\left(-\exp\left(-\frac{x-\xi}{\alpha}\right)\right) $$
The authors design a simulation to compare these estimators. On page 255, below Figure 4, why do the authors say
All the methods of estimation are invariant under linear transformations of the data, so without loss of generality the location and scale parameters $\xi=0$ and $\alpha=1$ were used throughout.
Here is the snap of the relevant page:
I think if we take other $\mu, \sigma$ values may also lead to different results for these estimators? Why does the author say that this is invariant?