R help - Oct 2010 - A subject related question

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?

warm thanks

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

Multivariate normal distributions with PSD variance matrices occur all the time.

The most common example in practice would probably be the distribution of the
vector of residuals from a normal regression.  It has a degenerate distribution
wrt R^n because it is subject to p linear restrictions.

Another common one is the variance matrix of the multinomial distribution, which
is singular since the frequencies are subject to a sum constraint \sum f = n. 
This is not a multivariate normal distribution, but becomes arbitrarily close to
one as the number of trials increases by the multivariate central limit theorem.

Bill Venables. 

-----Original Message-----
From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-project.org]
On Behalf Of Jonathan Beokhokhei
Sent: Sunday, 17 October 2010 4:17 AM
To: r-help at r-project.org
Subject: [R] A subject related question

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?

warm thanks

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