Dear Amir,

I don't know what kind of software you are using, but as Boby said, the scale matrix needs to be positive definite and therefore the software should be able to handle this. In R one can use the function rWishart() to simulate random matrices from a Wishart distribution. An example with an identity matrix as scale matrix would be

> rWishart(n = 1, df = 5, Sigma = diag(3))

, , 1

[,1] [,2] [,3]

[1,] 3.02230785 2.5532880 -0.03051006

[2,] 2.55328796 4.9008296 0.16212809

[3,] -0.03051006 0.1621281 0.91354357

> eigen(rWishart(n = 1, df = 5, Sigma = diag(3))[,, 1])

$values

[1] 16.826179 10.141205 1.328616

$vectors

[,1] [,2] [,3]

[1,] 0.6251332 -0.558390818 0.5453514

[2,] -0.6530738 0.008430981 0.7572473

[3,] 0.4274378 0.829535179 0.3593999

Regards, Stephan

Thank you Boby for your attention. I was thinking in the 2*2 matrix format and by "diagonal" I meant the other diagonal not the main one. I corrected the question. Thanks for your care.

I appreciate it if you can please give me your comments on this question now.

Thanks a lot Stephan. That was my mistake in asking the question. I corrected the question now.

My question is indeed about the identity matrix as the scale matrix. I can understand the essence of the identity matrix as the "scale" but I cannot understand the contradictory fact that what happens to the covariance of the variables that are correlated when sampling from a Wishart distribution with an I scale matrix.

Appreciate your comments here.

Thanks a lot

Amir

Dear Amir,

the Wishart distribution is constructed as the sum of the matrix product of  i.i.d. normally distributed random vectors Xi, which have the scale matrix Sigma as covariance matrix.  There are df of such products.

So if you choose Sigma to be the identity matrix, then the components of Xi will be independent. Further you will sample from a distribution, which has expected value equal to df * Sigma, i.e. the expected value is a matrix with zeros in the off-diagonal. But of course the probabilty of obtaining a realisation equal to the expected value will be zero. Hence each realisation will have non-zero off-diagonal elements.

A random vector with a covariance matrix due to a Wishart distribution, with the identity matrix as scale matrix, will have correlated components.

Regards, Stephan

Thanks a lot Stephan. It almost answers my question. I still have one small confusion though:

I have a random vector which is multivariate normally distributed with a covariance matrix (so as you said variables are correlated). Now, I want to have this covariance matrix sampled from a Wishart distribution with the I matrix as its scale.How come a none-zero covariance matrix will be sampled from a distribution with zero-off-diagonals ?

Dear Amir,

there are two things which you should have in mind

I hope this answers your question.

Regards, Stephan

Stephan,

Thank you very much for your reply. Yes that absolutely answers my question. Now I understand how a non-zero matrix can be sampled (or be a realisation) of a distribution with the mean of identity matrix.

Thanks a lot Stephan.

You simply generate Wishart matrices with identity matrix for the parameter matrix, and pre-multiply the generated matrix by the Cholesky factor of  the scale parameter matrix. This matrix follows a Wishart with the desired covariance matrix.