Talk:Fisher–Tippett–Gnedenko theorem

General comments

A suitable reference for the Theorem would be the book "Extreme value theory: an introduction" by Laurens de Haan and Ana Ferreira, published by Springer, 2006. The Fisher-Tippett-Gnedenko theorem is Theorem 1.1.3 of that book.

Tippett ends with a double t.

The second sentence of the article, saying "The maximum of a sample of iid random variables after proper renormalization converges in distribution to one of 3 possible distributions, the Gumbel distribution, the Fréchet distribution, or the Weibull distribution." is not true. For instance it is not true if the random variables come from a Poisson distribution or a Geometric distribution (Exercise 1.13 of the book cited), or from the distribution with CDF F(x)=1-exp(-x-sin(x)) on x>0 (Exercise 1.18). The exact Statement of the Theorem given in the article is fine, but to put it into words we need to express something corresponding to the hypothesis that the numbers a_n and b_n exist. So perhaps, "The maximum of a sample of n independent identically distributed random variables can sometimes be translated and rescaled to converge in distribution as n tends to infinity: when this happens, it converges to either the Gumbel, the Fréchet, or the Weibull distribution." Fathead99 (talk) 17:41, 15 February 2011 (UTC)

Thanks for the info. I've fixed the speling of Tipett, which is a start. I know nothing of extreme value theory myself, so I suggest you consider making the other changes you suggest yourself. Qwfp (talk) 19:09, 15 February 2011 (UTC)

Contributions

Can somebody please provide citations to the original works? In particular, I'd like to know what Gnedenko's contribution was on top of Tippett and Fisher's original formulae. Was it just the formalization of the results in terms of distribution theory? — Preceding unsigned comment added by 67.164.143.201 (talk) 19:24, 24 September 2013 (UTC)

Small 'wrong' detail in Haan's book

In the book they state that the class of limiting distributions is $G(ax+b)$ where

$G(x)=\exp(-(1+\gamma \,x)^{-1/\gamma })$

but in they proof, they reach:

$G(x)=\exp \left(-\left(1+\gamma \,{\frac {x-D(1)}{H'(0)}}\right)^{-1/\gamma }\right)$

Now place $b=D(1)$ and $a=H'(0)$. It is then more natural to say that the class will be $G((x-b)/a)$, with:

$G(x)=\exp(-(1+\gamma \,x)^{-1/\gamma })$

Can't really say the original is wrong, since one could do $a=1/H'(0)$ and $b=-D(1)/H'(0)$, and achieve the same statement as the one in the book.

I chose for the one I formulated, because it seems more intuitive to me, and it is the exact same "rescaling" used in the wiki article generalized extreme value distribution. Walwal20 (talk) 08:25, 15 July 2020 (UTC)

Disputed Section

The calculations about the normal distribution are simply incorrect. So I put a "disputed" tag there. The calculations simply do not make sense, for instance taking natural logarithm of the cdf does not output the claimed expression, etc. Someone should rewrite the correct asymptotics with an appropriate reference. — Preceding unsigned comment added by 2600:4040:2D1D:6C00:F048:DF83:977A:25E (talk) 01:48, 24 April 2023 (UTC)

Never mind, it seems ok but just written in an extremely convoluted fashion. Changed the "Disputed" to merely unreferenced. 2600:4040:2D1D:6C00:F048:DF83:977A:25E (talk) 01:54, 24 April 2023 (UTC)