[R] help computing a covariance

Eugene Salinas (R) r-eugenesalinas at comcast.net
Fri Jul 2 18:33:31 CEST 2004

Thanks. This is sort of what I have been trying to do... but I keep 
ending up with products of non-independent chi-squares where I am sort 
of getting stuck. I knew this Theorem just never knew it was called 
Cochran's Thm. Btw, do you know of a good book that deals with 
multivariate statistics using vector notation etc. All the books I have 
seem to be focused on scalar random variables and they don't even 
mention it.)

thanks, eugene.

Spencer Graves wrote:

>      Have you considered Cochran's theorem?  (A Google search just 
> produce 387 hits for this, the second of which 
> "http://mcs.une.edu.au/~stat354/notes/node37.html" provided details 
> that might help.)  By construction, P is n x n, idempotent of rank k, 
> so y'Py is chi-square(k).  Also, xA is an n-vector in the (rank k) 
> column space of x;  indeed, PxA = [x*inv(x'x)*x]xA = xA.  I can't see 
> the details now but I believe you can write (A'x'y)^2 = y'xAA'x'y as a 
> weighted sum of k independent chi-squares each with one degree of 
> freedom (since x and P have rank k), and then get what you want from 
> the sum of the weights.  Then check your result using Monte Carlo.
>      hope this helps.  spencer graves
> Eugene Salinas (R) wrote:
>> Hi everyone,
>> (This is related to my posting on chi-squared from a day ago. I have 
>> tried simulating this but I am still unable to calculate it 
>> analytically.)
>> Let y be an n times 1 vector of random normal variables mean zero 
>> variance 1 and x be an n times k vector of random normal variables 
>> mean zero variance 1. x and y are independent.
>> Then P is the projection matrix  P=x*inv(x'*x)*x'
>> I need to figure out the covariance
>> Cov ( y'*P*y , (A'*x'*y)^2 ) where A is a constant of dimension k 
>> times 1.
>> thanks, eugene.
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