Hypothesis Testing and the Generalised Extreme Value distribution Is it correct to say that the generality of the Fisher-Tippett theorem means block-maximum data will always fit a GEV distribution? And how can we reject hypotheses on GEV parameters?
Original question details:

GEV normally is used for block-maximum data, as per references like Coles: "An Introduction to Statistical Modeling of Extreme Values". I'm asking a general question about how hypotheses might be tested using the GEV given that it fits EV data from just about any source distribution.


This question was prematurely closed, so I'm reposting it.

 A: The Fisher-Tippett-Gnedenko theorem says that if there exist suitable rescaling sequences $a_n > 0$ and $b_n > 0$ such that
$$
\frac{\max\left(X_1, \ldots, X_n\right) - b_n}{a_n}
$$
has a non-degenerate limit distribution as $n \to \infty$, then the resulting limit distribution is the GEV (i.e., either Gumbel, Fréchet or Weibull).
It says nothing about the case where such sequences do not exist! Take, for example, the very heavy-tailed distribution $F_X(x) = 1 - \frac{1}{\log (x)}$. Then the fact that the expression above converges in law is equivalent to saying
$$
P\left(\frac{M_n - b_n}{a_n} \leq x\right) = F^{n}(a_n x + b_n) \to G(x)
$$
where $M_n = \max\left(X_1, \ldots, X_n\right)$ and $G(x)$ is the GEV distribution function.
But we notice that
$$
F^n(a_n x + b_n) = \left(1 - \frac{1}{\log(a_n x + b_n)}\right)^n =\left(1 - \frac{1}{\log(a_n x + b_n)}\right)^n = \exp\left( -\lim_{n \to \infty} \frac{n}{\log(a_n x + b_n)}\right),
$$
where the last equality holds only if the limit exists.
The limit in the last term will always be either $0$ or $+\infty$, which indicates that the limit distribution of the rescaled maximum is degenerate.
Both the FTG and Central Limit theorems propose limiting distributions for rescaled functionals, but both have necessary assumptions: for a $\mathrm{Student}(1)$-distributed sample, the rescaled mean will never be Normal, just as in the example above where the rescaled maximum will never be GEV.

To summarise some of the discussion in the comments, the OP is asking what the point is of testing goodness-of-fit for the GEV.
I would answer that the GEV is generally flexible enough for almost all observations (again, see example above) which are block maxima, but that it contains within it three distributions (Gumbel, Fréchet, Weibull) which are discriminated by the value of the shape parameter $\xi$. Therefore, testing the three cases of whether $\xi < 0$, $\xi = 0$ or $\xi > 0$ is something like a goodness-of-fit test of each of the three types, and therefore this should be (and is) the parameter of interest for hypothesis testing of the GEV.
