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Erroneous references

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The former statements about affine transformations are plainly false. They apply to Gaussian but *not* to Student t distributions. The two references that have this equation (https://www.researchgate.net/profile/B-M-Golam-Kibria/publication/267325829_A_short_review_of_multivariate_t-distribution/links/5fbd265e92851c933f56e29c/A-short-review-of-multivariate-t-distribution.pdf https://people.isy.liu.se/en/rt/roth/student.pdf) only state it, but it's wrong.

A=[2 1; 1 2], Sig=[1 0; 0 1] w/ Cauchy (\nu=1) shows a clear counter example.

I would strongly suggest to not include anything based on Kibria et al. and Roth unless it is clearly proven. Can somebody else please go through the other times that these works are referenced? (Multiple things are based on this.) — Preceding unsigned comment added by FelixPetersen (talkcontribs) 04:44, 26 September 2023 (UTC)[reply]

It was me who copied Kibria et. al. onto the Wikipedia entry relating to affine transforms. I'm not sure exactly what the issue is as I don't have a copy of of Kotz and Nadarajah's book to hand, where I believe they had a problem. However the affine transform pdf seems straightforward to prove using standard multivariate change of variable methods (in ~3 lines having looked) so what I would propose is that I write this proof up in lieu of other references and use it to replace the relevant paragraph section in the article.
I'll upload this to this location in the next couple of days for rechecking.
therustyone Therustyone (talk) 19:57, 16 October 2023 (UTC)[reply]
Affine Transform
Starting from a somewhat simplified version of the MV-t pdf: , where is a constant and is arbitrary but fixed, let be a positive definite matrix and form the transformed variable . Then, by a straightforward change of variables we get
The matrix of partial derivatives and the Jacobian becomes . Thus

In the denominator, we have which can be reduced to

where has replaced

Finally

which is a regular MV-t distribution.

Therustyone (talk) 09:56, 17 October 2023 (UTC)[reply]
The above is the latest iteration of a proof of the MV-t with affine transformation. Thus far, no issues with affine transforms have appeared and numerical integration of the transformed pdf's for p = 2 is consistent within numerical accuracy, ie. 10^-7. The transformed pdf works with symmetric transforms and also with orthogonal rotation matrices .
The next step, given no objections, is to tidy it up, submit for review, and hopefully return it to the main page. Therustyone (talk) 09:58, 18 October 2023 (UTC)[reply]

what is p

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What is p? Is it the dimension of ? Albmont 20:07, 6 March 2007 (UTC)[reply]

Added a comment to effect that p is the dimension Shaww 09:54, 19 April 2007 (UTC)[reply]

normalization

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The bivariate student distribution density function with zero correlation is presented as

But this does not normalize nicely to 1.

≠1

for any n. Only for high n it seems to go to 1.

Where is the (my) mistake?

jasper (talk) 17:06, 21 January 2008 (UTC)[reply]

A rough numerical integration with Matlab seem₫s to give an ok normalization to one:
>> p = inline('(1+(x^2+y^2)/n)^( -(n+2)/2 )/(2*pi)')
>> s=0; n=2; for x=-20:0.1:20, for y=-20:0.1:20, s=s+p(n,x,y);end,end,s
the result 99.5946 is approximately 100x0.1x0.1=1. fnielsen (talk) 10:12, 22 January 2008 (UTC)[reply]

Sorry, my mistake, fnielsen is right. My infinity was not big enough. Should I delete this entry in the discussion tab? jasper (talk) 16:51, 22 January 2008 (UTC)[reply]

I don't think you should delete; errors are good when we learn from them. Now, do the homework: prove analytically that the integral is one: replace and see what happens :-) Albmont (talk) 13:50, 1 February 2008 (UTC)[reply]

This article refers to elliptical distribution theory. I'd like to see a separate article about elliptical distributions. Duoduoduo (talk) 20:01, 11 March 2010 (UTC)[reply]

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Move discussion in progress

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There is a move discussion in progress on Talk:Student's t-distribution which affects this page. Please participate on that page and not in this talk page section. Thank you. —RMCD bot 04:01, 24 August 2021 (UTC)[reply]