Samples from a multivariate t distribution

Question

Hi I have the following problem. I draw a sample of a multivariate t-distribution with some fixed covariance matrix, so that the realizations are correlated, and $\nu=4$. Now I repeat this $n$ times, always with the same covariance matrix and the same $\nu$. I normalize each sample to mean 0 and standard deviation 1 and look at the histogram of all samples, it is a Gaussian distribution.

I somehow expected a t-distribution with the same degrees of freedom $\nu=4$. Why do I see a Gaussian distribution?

PS: I don't think the code is the problem since I already asked in a forum and it seems correct.

Thanks.

Answer 1

Your normalizing the samples to mean zero and variance one kills all of the heavy tail properties of the $t$-distribution. Since $T_\nu = z/v_\nu, z\sim N(0,1) \perp v_\nu \sim \chi^2_\nu$, your normalization of the scale by 1 annihilates $v_\nu$ forcing it to be one instead of a random quantity. The deviations from zero are due to the $z$, but you force them to have the variance of exactly one. As a matter of fact, I suspect that your normalization produces lower variance than one... probably 299/300 or 298/300, although you are not going to notice it on a histogram.

That the code runs (i.e., does not crash and does not produce errors) does not mean it is correct in the sense that it solves your problem... which you have not stated, anyway.

Answer 2

Here is an experiment that exhibits a clear lack of fit by a Gaussian density for a simulated t sample:

library(mvtnorm)
x=rmvt(1e4,sigma=matrix(c(1,.7,.7,1),nrow=2),df=4)
hist(x[,1]/sd(x[,1]),nclass=123,col="wheat2",prob=TRUE)
curve(dnorm,add=TRUE,lwd=2,col="sienna2")

However, since there are different definitions for the multivariate t, it may also be that you use one for which the marginals are not t's...

Answer 3

Because you get multivariate Student t as the fraction of the normal vector and a chi-squared random variable. So, of course, the histogram of entries of the vector will be normal, but for every simulation, it will be different normal, depending on what the chi-squared random variable shows you. If you normalize, chi-squared gets cancelled and it will just be standard normal all the time.