How understand "All the methods of estimation are invariant under linear transformations of the data"?

Question

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?

Answer 1

Imagine that the Gumbel distribution models the maximum temperature in a year. Would it matter if you use a Celsius temperature scale or a Fahrenheit temperature scale? No, the estimates will just shift and scale, but the properties, like efficiency, or relative variance ratios, remain the same.

If you perform simulations with a specific $\alpha = 1$ and $\xi = 0$ then the results will be relatively the same as when you would have performed simulations with other values (up to some shift and scaling).

Say you simulate the estimates $\hat{\xi}$ with some given true $\xi$ that produces variables $x$. If instead you would have made simulations with some different scale $x^\prime = a + bx$, then the estimate $\hat{\xi}^\prime$ for $x^\prime$ would be just a re-scaled value of the estimate $\hat{\xi}$ for the unscaled variables $x$.