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

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?

• Are you sure this paper uses MLE? I haven't read it, but its title and abstract focus on using moments for estimation. This is a location-scale family. It you're unsure what this means or implies, then please check out our posts on the topic.
– whuber
Nov 30, 2022 at 21:46
• Yes, on page 255. Nov 30, 2022 at 21:48
• Seriously?? The paper is behind a paywall.
– whuber
Nov 30, 2022 at 21:53
• The authors compare MLE and method of moments. But that's not the question I wanted to ask. I mainly want to ask why the authors only need to consider the special case of $\mu=0, \sigma=1$ (as true values) in the simulation and then calculate the MSE to compare which estimator performs well. How about other possible value of $\mu$ and $\sigma$? Nov 30, 2022 at 21:56
• That's what you seem to be asking in your previous question, so my recommendation is the same: learn about the meaning of location and scale.
– whuber
Nov 30, 2022 at 22:02

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 $$x$$.