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.

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    $\begingroup$ Could you explain what a "set of block-maximum data" means and describe the hypothesis you wish to test? $\endgroup$ – whuber Oct 19 '20 at 19:55
  • $\begingroup$ 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. $\endgroup$ – Isambard Kingdom Oct 20 '20 at 20:10
  • $\begingroup$ For the same reason that the Central Limit Theorem doesn’t cover all the possible distributions of the rescaled sample mean! First, the GEV is the limiting distribution if a suitable rescaling exists such that the rescaled block maxima converge, but this not always the case. Second, the rescaling identifies the specific location, scale and shape parameters of the limiting GEV distribution. $\endgroup$ – Maximilian Aigner Oct 20 '20 at 20:32
  • $\begingroup$ Maximilian,I'm not following your analogy with the CLT. Aren't there lots of distributions with cores that don't resemble a normal? $\endgroup$ – Isambard Kingdom Oct 20 '20 at 21:05
  • $\begingroup$ Maximilian, also, aren't the data that you note can't be suitably rescaled kind of unusual? It would help if I understood what kind of distributions give such data. An example in a book or published paper would help. Thank you. $\endgroup$ – Isambard Kingdom Oct 21 '20 at 4:08

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.

  • $\begingroup$ Thank you, Maximilian. Just so I understand this is not an answer to the question of what types of hypotheses might be tested. So that I'm communicating, if I fit a Gumbel to some data, a significance test allows me to reject or not reject a source distribution which is not regularly varying. If I fit a Frechet, then I can test for regular variation. If I fit a Weibul I test for regular variation at the domain endpoint. But all of these general properties are included in the GEV, hence I don't seem to be tesing for anything if I use the GEV. $\endgroup$ – Isambard Kingdom Oct 21 '20 at 18:59
  • $\begingroup$ @IsambardKingdom when you say that you are using a significance test, do you mean goodness-of-fit? E.g. after fitting by maximum likelihood? If you are talking about testing parameter values, the three types are subsumed under GEV and you certainly can test point values for location, scale and shape parameters. $\endgroup$ – Maximilian Aigner Oct 21 '20 at 20:10
  • $\begingroup$ The three choices you suggest correspond to testing different one-sided or point hypothesis on the shape parameter $\xi$, as far as I can tell. It remains the case that certain laws are neither slowly, or regularly varying, but have heavier tails (the case I used earlier is one such example, as the tail is as 1/$\log x$, heavier than any power law). $\endgroup$ – Maximilian Aigner Oct 21 '20 at 20:11
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    $\begingroup$ @IsambardKingdom often the real interest is in whether the tail has an upper limit (see e.g. the paper "Human life is unlimited – but short" by Rootzen and Zholud and its discussion) or whether certain moments exist; both reduce to tests on $\xi$. In that setting, you could consider $\mu$ and $\sigma$ as essentially nuisance parameters, and you would want the significance test to apply essentially to $\xi$. A confidence interval for $\xi$ gives you a lot of information: domain, whether moments exist, whether you should use Gumbel/Fréchet/Weibull.. The generality in the GEV lies there, IMO. $\endgroup$ – Maximilian Aigner Oct 21 '20 at 20:33
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    $\begingroup$ Okay, Maximilian, that is very helpful. I appreciate you having stuck with me through this exchange. It has helped me focus my ideas. Your note on establishing a confidence interval on $\xi$ is very helpful. I think I should allow this to be wrapped up, now. I still find the GEV to be odd, but for now, thank you. $\endgroup$ – Isambard Kingdom Oct 21 '20 at 20:37

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