Talk:Generalized extreme value distribution

Generalized Pareto Distribution
Why is the Generalized Pareto Distribution redirected to this article? First the Generalized Pareto Distribution is not the same as an Extreme Value Distriubution and second the Generalized Pareto Distribution is not mentioned a single time in the article.
 * I have changed the redirect, so it points now to Pareto distribution instead. DFH 18:52, 23 December 2006 (UTC)

edit --164.67.196.26 (talk) 23:20, 7 January 2008 (UTC)DBB

I believe the GPD is the same as the GEV using a change of variable so long as they are compared above the threshold used for the GPD. This can be found in eqn. 1.14 of "Extreme Values in Finance, Telecommunications, adnt eh Enviroment" edited by Barbel Finkenstadt and Holger Rootzen.

The distributions referred to as "Generalized Pareto" are not always the same. The distribution known in classic extreme vale theory (McNeil et al) as a GPD is referred to elsewhere as the "Extremal Pareto" (Kreps) which is the standard two-parameter Pareto with a redefinition of the shape parameter. Compare that with the Klugman-Panjer-Wilmot GPD. -- Avi (talk) 16:06, 14 October 2009 (UTC)

Question
what's the conditions on the sequences a_n and b_n?

Links to Frechet/etc
It's not completely obvious how the connection between Frechet and GEV distributions. It'll be really helpful to clean up those formulas - i.e., "mu" in the GEV distribution is not the same as "mu" in the frechet distribution. Probably similar issues with Gumbel/etc. — Preceding unsigned comment added by 74.88.101.86 (talk) 01:18, 23 September 2014 (UTC)

Disputed
The figure showing the Gumbel, Fréchet and Weibull is inconsistent with the equations. In particular, how is it that the Fréchet and Weibull  have different maximum heights? And why is it that the Weibull in the figure has a range that ends at 2?


 * I'm not familiar with these distributions, but I do see that the equations expressing them are the CUMULATIVE distribution functions, while the plots are of the probability DENSITY functions. If &mu; were equal to 2 (which should have been made clear in the plot) the Weibull distribution would end at 2. PAR 06:15, 16 January 2006 (UTC)


 * O.K., but compare to the specific article on the Weibull_distribution which says that x>0.


 * Right, the plots are not enough illustrative when the used parameters (esp. &mu;) are not presented.
 * To avoid even more confusion, let me say, that &mu; is not an expected value E(X) (mean that is commonly denoted &mu;) of the distribution here.
 * Instead, E(X) = 0 for all the 3 plotted distributions, while parameter &mu; = 2 for the Weibull-type distribution and some negative (near to -2?) for the Fréchet-type there.
 * Moreover, the presented Weibull-type distribution here is usually called reversed Weibull distribution because it is... reversed compared to Weibull distribution.
 * The linked article presents special form of the generalized Weibull distribution, where &mu; = 0, while this article refers to reversed form of the generalized Weibull distribution...
 * Thanks. (13 Mar 2006)

Figure shown and disputed is from the Matlab Statistics toolbox which fully describes the function definitions; see :
 * http://www.mathworks.com/access/helpdesk/help/toolbox/stats/bqem6hh-1.html
 * http://www.mathworks.com/products/statistics/demos.html?file=/products/demos/shipping/stats/gevdemo.html.


 * I have inserted "Reversed" and added some discussion. Melcombe (talk) 17:56, 21 February 2008 (UTC)

Provide practical guidance?
I arrived at this page with a specific, practical, and probably common, problem, and did not find the solution (or else did not recognize it). The problem is, if I believe a variable x follows some distribution, then what is the formula for the expected EVD of samples of size n? It would be very helpful to provide some examples along these lines for common distributions (eg Gaussian).

My best hint is to use a statistical evaluation program, e.g. the R package "extRemes" - it can give you parameters to a fit as well as goodness-of-fit estimations.

This might be obvious, but just in case, here goes: If the pdf of x is f(x) and it's cdf is F(x) and you have N samples, the exact pdf for the maximum from a sample of size N is Nf(x)F(x)^(N-1).

[There seems to be a misconception here. The EVD provided applies to all probability distributions asymptotically (i.e., as the number of samples increases without limit) much as the Gaussian distribution is the asymptotical limit for N summed variables of any distribution. Obviously, different underlying distributions will have different values for the parameters. If F(x) is any cumulative distribution function, the probability that some X is the maximum of N values is given by F(X) raised to the Nth power. Letting N increase without limit, one can derive the EVD distributions.] —Preceding unsigned comment added by Jymcr (talk • contribs) 17:24, 4 September 2007 (UTC)
 * That looks correct. 69.143.236.33 00:28, 18 October 2007 (UTC)

References with the answers you are looking for
Some References:

- To understand the three (Gumbel, Weibull and Frechet) page 7 and 8: http://www.gloriamundi.org/picsresources/pesrgs.pdf

- An Introduction to Statistical Modeling of Extreme Values; By Stuart Coles http://books.google.com/books?id=2nugUEaKqFEC&pg=PA1&dq=introduction+extreme+value&sig=7kZzjXGcEo7iNXDxOuUkGm_3LP0#PPA2,M1

-Extreme Values in Finance, Telecommunications, and the Environment; By Bärbel Finkenstädt, Holger Rootzén http://books.google.com/books?hl=en&lr=&id=d55hUIBERf0C&oi=fnd&pg=PR11&dq=Generalized+Pareto+versus+Generalized+Extreme+Value&ots=mh0R8Vk1Wn&sig=TZla5IGDMRiCDJ5HLCoJVlfC4g4#PPA79,M1

Andrea Graves, 10 September 2007, London, UK. —Preceding unsigned comment added by Andrea.graves (talk • contribs) 00:29, 10 September 2007 (UTC)

Some updates needed
I think if xi < -1 then the formula currently stated for the mode is not correct. I think when xi <-1 the mode is at the upper limit of the distribution, mu-sigma/xi. Also I think the formulae currently stated for the mean, variance, skewness and kurtosis are only valid if xi < 1, 1/2, 1/3 and 1/4 respectively. Also quite a few of the formulae stated in the table need to have a separate formula given for the special case xi=0 (in the same way as is currently done for the support).Fathead99 (talk) 10:17, 3 October 2008 (UTC) Also I think the numerator of the skewness should be (g_3-3g_1g_2+2g_1^3)*sign(xi). (So the current form is right only when xi<0.) Fathead99 (talk) 15:15, 3 October 2008 (UTC)

This link (currently Note #6) was broken:. I replaced it with this, which I think is correct but not sure:. Jess_Riedel (talk) 22:15, 4 December 2019 (UTC)

Corrected parameter order in Weibull distributions
In the section Related distributions, I had to change the order of the parameters for the Weibull distribution to be consistent with the article on the Weibull distribution to which this article must surely give deference on such matters. It gives scale parameter first, then the shape parameter. (This is the opposite, by the way, of the convention used by R_(programming_language) in its functions rweibull, etc.)

Scwarebang (talk) 22:54, 24 November 2013 (UTC)

The limit "ksi = 0
Looks like it is exp(-(x-m)/s) exp(-exp(-(x-m)/s)) to me and not exp(-(x-m)/s). Am I wrong ? — Preceding unsigned comment added by 87.202.21.225 (talk) 05:26, 18 December 2013 (UTC)

Removal of unknown parameter from the probability distribution template
There was a parameter causing the problem Warning: Page using Template:Infobox probability distribution with unknown parameter "g_k" (this message is shown only in preview).

Unfortunately I do not know what that parameter means, so I am recording it below in case someone knows about it and would like to place in the article:

| g_k       = $$g_k=\Gamma(1-k\xi)$$

rendered as: g_k = $$g_k=\Gamma(1-k\xi)$$ — Preceding unsigned comment added by Walwal20 (talk • contribs) 17:05, 20 May 2020 (UTC)

Parameters of the minimal GEV
For completeness, I think it would be useful to add a line to section 3.1 pointing out that the formulas of the mean, median, variance... of the minimal GEV are the same as for the maximal GEV (as one can see here: https://reference.wolfram.com/language/ref/MinStableDistribution.html) — Preceding unsigned comment added by 2001:8A0:5745:EB01:C1B7:9C5A:3EAF:5E79 (talk) 08:40, 10 July 2021 (UTC)

Example of normal distribution unclear
The example of the maximum of N iiv following the normal distribution is not clear enough in its current state. What is the mentioned Φ function? The paragraph refers to the article Fisher–Tippett–Gnedenko theorem, but nothing in the latter helps understanding the claimed statement. Indeed, there is also a discussion in the latter article about the maximum of normally distributed variables, but it is quite informal and in any case, it is not clear at all how both explanations are linked.

Moreover, since no details / rigorous proofs are given, a reference would be welcome. Infisxc (talk) 13:43, 14 September 2023 (UTC)