# wishart distribution and correlation matrix???

#### will22

##### New Member
hey there everyone! just a quick question....

you see how the sampling distribution of the covariance matrix follows a wishart distribution (for the multivariate normal case)?

does it make sense to say then that the sampling distribution of the correlation matrix also follows a wishart distribution? it makes sense to me because i look at correlation matrices as scaled covariance matrices, but i need this piece of info for a little "debate" i'm having with my advisor and i cant find any sources who specifically talk about the correlation matrix.

just wanna verify this with someone who's a little bit more well-versed in these topics... a reference (book, journal article, etc...) would be great! thanks!

#### Dason

I'm going to say no. Correlation is bound between -1 and 1 so the support isn't over all positive definite matrices once you convert to correlation matrices.

I don't know though... A wishart might make a reasonable approximation if the parameters are set right but in general I would say the correlation matrix can't follow a wishart exactly for the reasons given.

#### will22

##### New Member
I don't know though... A wishart might make a reasonable approximation if the parameters are set right.
well, thing is i've been reading a little bit on the topic and it says that the sample covariance matrix for a sample from a multivariate normal distribution follows the wishart distribution. so my reasoning was like this: if we take a standard normal distribution then it has a 0-mean vector and it's covariance matrix would be a correlation matrix. so i guess when you said "the parameters are set right" is because this will only happen in the case of the standard multivariate normal... right? no? maybe?

#### Dason

well, thing is i've been reading a little bit on the topic and it says that the sample covariance matrix for a sample from a multivariate normal distribution follows the wishart distribution. so my reasoning was like this: if we take a standard normal distribution then it has a 0-mean vector and it's covariance matrix would be a correlation matrix. so i guess when you said "the parameters are set right" is because this will only happen in the case of the standard multivariate normal... right? no? maybe?
But the sample covariance matrix doesn't need to be a correlation matrix. In the sample it's possible (although highly unlikely) to see a covariance matrix with really large values even if you sampled from a standard normal. Now as the sample size increases these will decrease.

Just because the population covariance matrix is a 'correlation matrix' doesn't mean that the sample covariance matrices will still have any of those properties.

#### will22

##### New Member
Just because the population covariance matrix is a 'correlation matrix' doesn't mean that the sample covariance matrices will still have any of those properties.
gotcha... i think you're pointing me on the right direction here. it is generally not the case that the sample correlation matrix will follow a wishart distribution. i just need to find out under which conditions it does.

at least now i know where to start.
thanks!

#### will22

##### New Member
oh wow... you're beyond awesome. it looks like a great place to start looking at this.
thanks!

#### Lep

##### New Member
A white wishart matrix will be based on covariance between columns of n by p data matrix X. If you use random standard normal variants (mean zero variance one) the scale of all the variables will be the same. Given this, the variances on the diagonal of the corresponding covariance matrix should approach one as the number of rows in X tend to infinity. Here there won't be much difference between core and cov since the variance of a variable is one.