R help - Sep 2003 - what does the sum of square of Gaussian RVs with differen t variance obey?

This is a relatively recent article that is somewhat accessible.
Jensen, D. R., and Solomon, Herbert (1994), "Approximations to joint
distributions of definite quadratic forms", Journal of the American
Statistical Association, 89 , 480-486
It has references to previous work.

I also have an old paper that is so old I can't tell what journal it came
out of:(
Grad, Arthur and Solomon, Herbert "Distribution of Quadratic Forms and Some
Applications" probably published in 55 or 56 but I can't tell.  The
paper by
Grad and Solomon uses the moment generating function to give the exact
distribution and various approximations to produce a table for a sum of 2 or
3 variates.


Usual disclaimers ...
Bob


-----Original Message-----
From: Thomas Lumley [mailto:tlumley@u.washington.edu] 
Sent: Tuesday, September 23, 2003 10:07 AM
To: Jean Sun
Cc: r-help@stat.math.ethz.ch
Subject: Re: [R] what does the sum of square of Gaussian RVs with different
variance obey?

On Tue, 23 Sep 2003, Jean Sun wrote:
> >From basic statistics principle,we know,given several i.i.d Gaussian
> >RVs with zero or nonzero mean,the sum of square of them is a central or
> >noncentral Chi-distributed RV.However if these Gaussian RVs have
> >different variances,what does the sum of square of them obey?
>
Nothing very useful.  It's a mixture of chisquare(1) variables. One
standard approach is to approximate it by a multiple of a chisquared
distribution that has the correct mean and variance.

	-thomas

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RBaskin at ahrq.gov writes:
> This is a relatively recent article that is somewhat accessible.
> Jensen, D. R., and Solomon, Herbert (1994), "Approximations to joint
> distributions of definite quadratic forms", Journal of the American
> Statistical Association, 89 , 480-486
> It has references to previous work.
> 
> I also have an old paper that is so old I can't tell what journal it
came
> out of:(
> Grad, Arthur and Solomon, Herbert "Distribution of Quadratic Forms and
Some
> Applications" probably published in 55 or 56 but I can't tell. 
The paper by
> Grad and Solomon uses the moment generating function to give the exact
> distribution and various approximations to produce a table for a sum of 2
or
> 3 variates.
Looks like this one (courtesy of JSTOR):

    Distribution of Quadratic Forms and Some Applications

        Arthur Grad; Herbert Solomon

        The Annals of Mathematical Statistics, Vol. 26, No. 3. (Sep.,
        1955), pp. 464-477.



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