Re: Multivariate student t: normalization in mvtrnd

[Arno Onken]

Hi Iñigo,

On 11/28/12 17:11, Iñigo Urteaga wrote:
>> Well, it's not hard to rescale the result of mvtrnd to get an arbitrary
>> covariance matrix. You might also want to add a constant to get a
>> non-zero mean. In contrast to mvnrnd, mvtrnd doesn't have arguments for
>> that.
> 
> Yes, adding a non-zero mean is trivial, but how would you rescale the
> result of mvtrnd to get an arbitrary covariance matrix? That is, if a
> a covariance matrix V is provided, which is normalized by mvtrnd to
> the correlation matrix R=V./(sqrt(diag(V)*diag(V))), then what is the
> resulting covariance of the obtained samples in terms of V?

The resulting covariance of the samples in terms of V is
nu/(nu-2)*V./(sqrt(diag(V)*diag(V))). So the scaling

samples = mvtrnd (V, nu, M) * diag (sqrt (diag (V));

should do the trick - correct me if I'm wrong. That's the same
"correlation to covariance" scaling that you would use for the
multivariate normal if you had generated samples with a correlation matrix.

>> mvtrnd just gives you the equivalent of the standard normal.
> 
> As far I understand it, mvtrnd provides the standard student t
> distribution only when the provided correlation matrix R is the
> Identity matrix. For the rest of cases when R is not the identity
> matrix, the resulting samples are not distributed according to the
> standard t, am I right?

I didn't mean it that literal. I just wanted to point out a conceptual
similarity in terms of parametrization. You are right. The standard
multivariate student t distribution is typically defined with R the
identity matrix. mvtrnd doesn't provide the standard multivariate
student t distribution if R is not the identity matrix (the marginals,
however, will still be standard student t distributed). So it's
conceptually similar in terms of mean and variance but not in terms of
covariance.

On a side note, there are actually cases in which it's nice to have the
parametrization that mvtrnd has. For instance, if you want to generate
student t copula samples, then you need exactly the parameter space that
mvtrnd is designed for.

Arno