In the question https://stackoverflow.com/questions/18153450/generating-random-variables-from-the-multivariate-t-distribution, I am confused by why the answer requires we modify the Sigma in such a way that we need to multiply is by (D-2)/D. Here sigma is the covariance matrix for me. The answer also mentions that the correlation matrix is defined when df > 2, shouldn't it be df>= 2? This is because the correlation coefficient can't be calculated when the data is continue itself one, there must be more than 1 series. Am I interpreting this correctly??
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1$\begingroup$ With 1 degree of freedom, the T-distribution (Cauchy) has divergent 1st moment (and all subsequent moments). With only 2 degrees of freedom, the T-distribution has divergent 2nd moment (and all subsequent moments). So the variance of a T-distribution is undefined for degrees of freedom less than 3. As we know the scale of a T-distribution is not the covariance, as with normal distributions. If you think of how the degrees of freedom play into estimation of the covariance of T distributed RVs, it makes sense to calculate the scale matrix as such. $\endgroup$– AdamOCommented Oct 23, 2013 at 23:29
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$\begingroup$ That ratio is used for the same reason it comes up in the univariate $t$ (which is that the variance of the standard $t$ is $\frac{\nu}{\nu-2}$. And no, it's $>$ not $\geq$ (what's $\frac{\nu}{\nu-2}$ when $\nu=2$?). $\endgroup$– Glen_bCommented Oct 24, 2013 at 0:37
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