What motivates, justifies, gives rise to the differences between the Gumbel, Fréchet, and Weibull distributions? Glen_b's comment indicates that they are distributions for extreme values generated by different kinds of distributions in IID sequences. What are those different kinds? [EDITed in response to Glen_b's comment.]

Feel free to point me to other sources.

[Background: I'm just beginning to learn a little bit of extreme value theory (mostly by working through early sections Reiss & Thomas's Statistical Analysis of Extreme Values 2nd ed.). I understand the mathematical definitions of the fully parameterized Gumbel, Fréchet, and Weibull distributions, how they can all be viewed as special cases of a generalized Extreme Value distribution, and that for some parameter combinations a Fréchet or Weibull distribution can be close to a Gumbel distribution. I'm trying to get insight into these distributions' differences at a more conceptual level.]

[I have no reason to doubt that Glen_b is right, but just in case, here are parts of the original question that no longer seem relevant: Are there real-world examples or mathematical contexts that help to motivate, for example, the use of a Fréchet distribution as opposed to a Gumbel distribution? (cf. radioactive particle emissions as motivation for the Poisson distribution.) Or might it be useful to see the three distributions as derived from some other distributions, perhaps as a limit as some parameters go to 0 or $\infty$? (cf. Poisson distribution as a limit of a binomial distribution.) A closely related question: Why would you choose to model extreme values with one of the three distributions rather than the other two (other than simply thinking that the data look like they'd fit one distribution better)?]

  • 1
    $\begingroup$ The three types don't arise by human choice (so 'motivation' is the wrong word). That they occur is an observation about how extreme values behave - different original distributions lead to the extreme values having one of the three types of distribution. As in 'let's work out what the extreme value distribution can look like'... 'oh, look, it's always one of the types A, B or C' $\endgroup$
    – Glen_b
    Feb 7, 2014 at 23:56
  • $\begingroup$ Thanks @Glen_b. That's very helpful. I've edited question accordingly. (I meant "motivate" in a broad sense that's consistent with your remark; I had to use a vague term because I didn't know what more-specific question to ask.) $\endgroup$
    – Mars
    Feb 8, 2014 at 3:26
  • 1
    $\begingroup$ You might find the discussion here useful. Specifically, as it says in the first paragraph "By the extreme value theorem the GEV distribution is the only possible limit distribution of properly normalized maxima of a sequence of independent and identically distributed random variables" ... that second link is to the Fisher–Tippett–Gnedenko theorem, the mathematical justification for that statement. $\endgroup$
    – Glen_b
    Feb 8, 2014 at 3:39
  • $\begingroup$ Ah, good. The Fisher–Tippett–Gnedenko Wikipedia page doesn't give me all of the insight I'd hoped for, but it's a start, and is helping me to search in other resources. Hadn't noticed it. (Reiss & Thomas has the easiest presentation, for me, that I've found, but the presentation of F-T-G is spread out and not really highlighted; now that I know what to look for, I'm finding its components.) $\endgroup$
    – Mars
    Feb 8, 2014 at 18:29
  • 1
    $\begingroup$ Gnedenko's original paper, which is republished in Kotz & Johnson Breakthroughs in Statistics Volume I, uses elementary techniques and provides ample insight. $\endgroup$
    – whuber
    Feb 9, 2014 at 19:27


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Browse other questions tagged or ask your own question.