Talk:Generalized normal distribution

Fat Error

The PDF formula presented for the generalized normal distribution version 1, aka exponential power distribution aka generalized error distribution, is NOT a generalized NORMAL distribution, but most probably a generalized Laplace distribution. The PDF formula presented cannot reproduce the unmodified normal distribution! alpha=1 leads to sd= 0.707 (Normal distribution: sd=1) and alpha= sqrt(2) leads to density(0)= 0.2829 (Normal Distribution: density(0)= 0.3989). In case of doubt, I can prove the R code to reproduce my calculations. — Preceding unsigned comment added by Consuli74 (talk • contribs) 09:15, 29 June 2016 (UTC)

All this is correct! A better kind of symmetrical generalized Gaussian or normal distribution would be the Student-t as normal/chi(df). The only disadvantage is that the Student-t cannot model short tails. This would be possible by using e.g. uniform / chi(df). When we now create a distribution which changes from uniform to normal with a parameter p, then we end up in a general symmetrical Gaussian with two parameters p and df. This approach needs even no inventions because such switch is just the (normalized) Irwin-Hall distribution IH(p), with p=1 we get the uniform, with p=2 the triangular, and with infinite p we can approach the normal distribution. The only disadvantage is that p would be discrete, but solving this is no problem, as for df. — Preceding unsigned comment added by 185.217.253.59 (talk) 13:12, 9 August 2018 (UTC)

Merger proposal

This is the same distribution as Generalized normal distribution. While it shouldn't be too hard to merge the two, the main question is which name to use? Any comments? -3mta3 (talk) 12:46, 2 March 2009 (UTC)

One thing that needs thinking about is how to handle other distributions which are also called "generalised normal". For example, documentaion for R includes a generalised normal distribution which is not the one referred to here (see eg. [1] ). Nether of these distributions is what is called either the skew-generalised normal or skew-normal distribution. Also I note that the ISI glossary has "Kapteyn's univariate distribution" as an alternative name for a generalised normal (not clear which) [2], but I guess this name should be avoided. Melcombe (talk) 14:31, 2 March 2009 (UTC)

I agree. I added this page recently because I didn't find this distribution (under either name) on the List of probability distributions page. Since "Gaussian distribution" redirects to "Normal distribution", I propose that we merge these two pages under "generalized normal distribution" with a redirect from "generalized Gaussian distribution." Then we can add a comment in the text about the "Kapteyn" name. I don't know what to do about the other generalized normal distribution. Maybe we could have them on the same page with two copies of the probability distribution template. Skbkekas (talk) 16:39, 2 March 2009 (UTC)

On further investigation, the generalized normal distribution referred to in the R documentation cited above ([3]) does not appear to include the normal distribution as a special case (also, the literature reference in the R code is to a 1990 paper of Hosking's, but this paper does not discuss anything like the distribution in the R code, and I didn't find any use of the term "generalized normal" in Hosking's other papers on JSTOR). In that sense, it is "generalized" in the same way that the lognormal, inverse normal, and half-normal distributions are (i.e. derived from the normal via a transformation). I think we can clarify that "generalized here" means a parameteric family that includes the normal distribution as a special case. This includes skew-normal, which already has a page that we can link to. Skbkekas (talk) 19:22, 2 March 2009 (UTC)

The above was moved from Talk:Generalized Gaussian distribution. Melcombe (talk) 10:34, 3 March 2009 (UTC)

Actually the generalized normal of Hosking does include the normal distribution as a special case, as well as the three-parameter log-normal of both positive and negative skewness. In one sense it might really be only be a form of reparameterisation of the three-parameter lognormal distribution, but the point here is that it appears in the literature under the name "generalized normal". Melcombe (talk) 10:46, 3 March 2009 (UTC)

Given that Hosking's version is in most cases just a log-normal, it might be better overall to have a separate article for that parameterisation but calling it something like "three-parameter lognormal" or "shifted lognormal", or even "generalised log-normal". I think putting several diffeent families of distributions in a single article is too confusing. The diagrams are an excellent contribution though. Melcombe (talk) 09:45, 5 March 2009 (UTC)

Melcombe is correct that these are both generalizations of the normal distribution in the same sense. I have included a discussion of both of them on this page, calling them "version 1" and "version 2" for lack of better terms. I don't have any references for "version 2" except the R documentation, so if someone could add the appropriate reference to Hosking's book or paper that would be great. Skbkekas (talk) 15:37, 5 March 2009 (UTC)

I have added a reference for this. However, it seems that by the time of that book they had decided to changec the name to just "lognormal distribution" and it appears only under that name in the book. However, it is certainly the same distribution as Hosking originally called the "generalized normal" . The earliest ref is "Hosking, J. R. M.: 1986, The theory of probability weighted moments. IBM Research Report, RC12210" but this is not readiliy accessible. The "generalized normal" terminology has been used by others based on this earlier report: for example, from 1998, http://www.springerlink.com/content/pk6871x147547766/ , and, from 2007, http://linkinghub.elsevier.com/retrieve/pii/S0022169407005069 . Melcombe (talk) 12:34, 6 March 2009 (UTC)

Multivariate Version

We need a multivariate generalization for this function. Currently, only the univariate version is given (where x and alpha are scalars). I was almost able to figure out the multivariate version (where x is a vector and alpha is a matrix), but I couldn't figure out the scale factor (to ensure unit variance).

Almon.David.Ing (talk) 17:22, 17 June 2009 (UTC)

Version 1 Questions

Does the claim of continuous derivatives under "Parameter Estimation" refer to only to the \beta? The LaPlace distribution (\beta=1) has no derivative in \mu at zero, which contradicts the claim of floor(\beta)=1 continuous derivatives in the text. Similarly, I think that the loglikelihood is infinitely smooth is \alpha. Some clarification would be nice. Also, I don't understand the CDF plot; since all of the exponential power densities as illustrated have \mu=0, the CDF of the \beta=0.5 case should be 0.5 when x=0 (like all of the others)...? Actually, the CDF of the \beta=1 case is wrong too; there must be a bug in the integrating function used for the plots. I looked at the python code and couldn't find it, unless scipy's gammainc function is buggy which would be odd.

69.201.131.239 (talk) 04:36, 19 March 2012 (UTC)

The CDF of version 1 seems to have a value other than zero at x = -\inf which is odd. It works out correct if the \Gamma(1/\beta) is not present in the denominator of the second term. — Preceding unsigned comment added by 27.251.48.50 (talk) 06:13, 28 March 2012 (UTC)

ah, okay. python must use a normalized incomplete gamma. i would redo the graphs but i have a mac and right now i don't feel like jumping through the hoops to install numpy and scipy.

The CDF plot is indeed wrong (as pointed out, since the PDFs are even CDFs should all equal 0.5 at x = 0); it seems like it just shows the integral of the PDFs as shown in the figure above, i.e. on the interval [-3, 3]. Would anyone who's good with Python be able to fix this? — Preceding unsigned comment added by 151.225.20.72 (talk) 08:19, 25 July 2013 (UTC)

Fixed the CDFs. The plot is visually accurate when simply re-running the original code in a modern python setup. jugander (t) 00:44, 28 April 2015 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Generalized normal distribution/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

This article addresses the univariate function (where x and alpha are univariate). It could be expanded to address the multivariate function (where x is a vector and alpha is expressed as a matrix). I was almost able to derive this, but I am not sure about the scale factor (to ensure unit area). Almon.David.Ing (talk) 17:12, 17 June 2009 (UTC)

Last edited at 17:12, 17 June 2009 (UTC). Substituted at 03:11, 3 May 2016 (UTC)

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Version 2 issues

I've been trying to use the PDF that's listed for version 2, but it doesn't seem to work. In order to better understand the PDF I tried to follow the links (11 in particular since it's online, but it's broken). I was able to Google the documentation in question, but it does not give details on what the actual function in R does (or what formula it's following). The details listed for 'version' 2 are pretty sparse and I've spent well over an hour at this point trying to make the function work as listed (and to track down more details on it, short of trying to get the book listed through an interlibrary loan).

The PDF listed suggests that it is proportional to Φ(y)/x, where Φ(y)= 1/sqrt(2) exp(-y^2). However, it then suggests that y (when κ≠0) is -1/κ...where the negative sign would be counteracted by the square...?

The log function (I assume base 10 log?) leads to imaginary values in the exponential, which leads to large oscillations/asymptotes in the final plot.

Anyway, I'm not sure what the right answer is, but Version 2 section could use a bit of clarification. This seemed like the best place to bring the subject up. — Preceding unsigned comment added by Plasma geek (talk • contribs) 16:36, 29 June 2017 (UTC)

Renaming sections

"Version 1" and "Version 2" are quite arbitrary; I'd like to propose renaming them to Symmetric and Asymmetric. fgnievinski (talk) 01:08, 22 June 2022 (UTC)