Talk:Multivariate t-distribution

Erroneous references

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 (talk • contribs) 04:44, 26 September 2023 (UTC)

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)

Affine Transform

Starting from a somewhat simplified version of the MV-t pdf: $f(X)={\frac {\mathrm {K} }{\left|\Sigma \right|^{1/2}}}\left(1+\nu ^{-1}X^{T}\Sigma ^{-1}X\right)^{-\left(\nu +p\right)/2}$, where $\mathrm {K}$ is a constant and $\nu$ is arbitrary but fixed, let $\Theta$ be a positive definite matrix and form the transformed variable $Y=\Theta X$. Then, by a straightforward change of variables we get

$f_{Y}(Y)={\frac {\mathrm {K} }{\left|\Sigma \right|^{1/2}}}\left(1+\nu ^{-1}Y^{T}\Theta ^{-T}\Sigma ^{-1}\Theta ^{-1}Y\right)^{-\left(\nu +p\right)/2}\left|{\frac {\partial Y}{\partial X}}\right|^{-1}$

The matrix of partial derivatives ${\frac {\partial Y_{i}}{\partial X_{j}}}=\Theta _{i,j}$ and the Jacobian becomes $\left|{\frac {\partial Y}{\partial X}}\right|=\left|\Theta \right|$. Thus

$f_{Y}(Y)={\frac {\mathrm {K} }{\left|\Sigma \right|^{1/2}\left|\Theta \right|}}\left(1+\nu ^{-1}Y^{T}\Theta ^{-T}\Sigma ^{-1}\Theta ^{-1}Y\right)^{-\left(\nu +p\right)/2}$

In the denominator, we have $\left|\Sigma \right|^{1/2}\left|\Theta \right|$ which can be reduced to

$\left|\Sigma \right|^{1/2}\left|\Theta \cdot \Theta \right|^{1/2}=\left|\Theta \Sigma \Theta ^{T}\right|^{1/2}$

where $\left|\Theta \right|$ has replaced $\left|\Theta ^{T}\right|$

Finally

$f_{Y}(Y)={\frac {\mathrm {K} }{\left|\Theta \Sigma \Theta ^{T}\right|^{1/2}}}\left(1+\nu ^{-1}Y^{T}\Theta ^{-T}\Sigma ^{-1}\Theta ^{-1}Y\right)^{-\left(\nu +p\right)/2}$

which is a regular MV-t distribution.

Therustyone (talk) 09:56, 17 October 2023 (UTC)

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 $\Theta$ transforms and also with orthogonal rotation matrices $\Theta =[cos\phi ,sin\phi ;-sin\phi ,cos\phi ]$.

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)

what is p

What is p? Is it the dimension of $\Sigma$? Albmont 20:07, 6 March 2007 (UTC)

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

normalization

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

$f(t_{i})={\frac {1}{2\pi }}(1+(t_{1}^{2}+t_{2}^{2})/n)^{-(n+2)/2}.$

But this does not normalize nicely to 1.

$\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f(t_{1},t_{2})\,dt_{1}\,dt_{2}$≠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)

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)

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)

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 $t1=r\cos \theta ,t2=r\sin \theta \,$ and see what happens :-) Albmont (talk) 13:50, 1 February 2008 (UTC)

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)

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