[R] A subject related question

[Duncan Murdoch]

On 16/10/2010 2:17 PM, Jonathan Beokhokhei wrote:
> Dear friends, please allow me a naive subject oriented question at this moment. I was wondering whether VCV matrix for some multivariate normal distribution can be PSD (which I always thought must be PD).
>
> I came across that point as I was working on some sample distribution of some statistic which involves population correlation matrix. As correlation matrix always a PSD, it seems that that sample distribution (that is asymptotically normal) comes with some vcv matrix which is PSD.
>
> Can somebody help me to sort this out?

I suppose it depends on your definitions, but the following gives a 
bivariate normal with a singular covariance matrix:

X1 ~ N(0,1)
Y1 = X1

Then the pair (X1, Y1) is bivariate normal and singular.  It has no 
density with respect to Lebesgue measure on R^2.

Duncan Murdoch