R help - Mar 2007 - covariance question which has nothing to do with R

This is a covariance calculation question so nothing to do with R but
maybe someone could help me anyway.

Suppose, I have two random variables X and Y whose means are both known
to be zero and I want to get an estimate of their covariance.

I have n sample pairs 

(X1,Y1)
(X2,Y2)
.
.
.
.
.
(Xn,Yn)

, so that the covariance estimate is clearly 1/n *(sum from i = 1 to n
of ( X_i*Y_i) ) 

But, suppose that it is know that the X_i are positively correlated with
each other and that the Y_i are independent
of each other.

Then, does this change the formula for the covariance estimate at all ?
Intuitively, I would think that, if the X_i's are positively
correlated , then something should change because there is less info
there than if they were independent but i'm not sure what should change
and I couldn't find it in a book.  

I can assume that the correlation between the X_i's is rho if this makes
things easier ? Thanks.

References are appreciated also.
--------------------------------------------------------

This is not an offer (or solicitation of an offer) to buy/se...{{dropped}}

while my other program is running.....

The reference I mentioned previously addresses exactly this. Snijders and 
Bosker's Multilevel Analysis book on page 31 and 33, section 3.6.2 and 363 
discuss this.

When you say that the Xs are correlated then you would need to say according to 
which structure they are correlated:


(1,X1,Y1)
(1,X2,Y2)
(1,X3,Y3)
.
.
.
(1,X55,Y55)
(2,X56,Y56)
(2,X57,Y57)
.
.
.
(2,...................

To pick some real world examples one row represents a person, or a stock. And 
the first column indicates to which organization or to which country that 
person/stock belongs to. Then Xs are correlated within the organization/country.
You will have two covariances, one within-county and one between country 
covariance of stocks.
This can be implemented in R manually providing method of moments estimates, or 
the gls function providing ML or REML estimates can be used for that.

I am not a post doc, just a pre master :-)

Toby


Leeds, Mark (IED) wrote:
> This is a covariance calculation question so nothing to do with R but
> maybe someone could help me anyway.
> 
> Suppose, I have two random variables X and Y whose means are both known
> to be zero and I want to get an estimate of their covariance.
> 
> I have n sample pairs 
> 
> (X1,Y1)
> (X2,Y2)
> .
> .
> .
> .
> .
> (Xn,Yn)
> 
> , so that the covariance estimate is clearly 1/n *(sum from i = 1 to n
> of ( X_i*Y_i) ) 
> 
> But, suppose that it is know that the X_i are positively correlated with
> each other and that the Y_i are independent
> of each other.
> 
> Then, does this change the formula for the covariance estimate at all ?
> Intuitively, I would think that, if the X_i's are positively
> correlated , then something should change because there is less info
> there than if they were independent but i'm not sure what should change
> and I couldn't find it in a book.  
> 
> I can assume that the correlation between the X_i's is rho if this
makes
> things easier ? Thanks.
> 
> References are appreciated also.
> --------------------------------------------------------
> 
> This is not an offer (or solicitation of an offer) to buy/se...{{dropped}}
> 
> ______________________________________________
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