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st: RE: RE: RE: RE: Wishart distribution


From   "Naji Nassar \(MIReS\)" <naji.nassar@mires.fr>
To   <statalist@hsphsun2.harvard.edu>
Subject   st: RE: RE: RE: RE: Wishart distribution
Date   Fri, 16 Apr 2004 10:09:18 +0200

Dear listers

Sorry if this is unrelated to Stata.


> The only way you could formulate such a test would be if you have a
(large)
> sample of matrices.

That's the case, I'll have 200 covariance matrix (simulation case, same
sample size, same theoretical covariance matrix).
The original data are simulated, non normal (Ramberg method), 200
simulations for each case

The context :
In causal modelling, ML is efficient under the hypothesis that data is
multivariate normal distributed. But even when the data are not normal,
covariance method is still accurate (intermediate results) in some cases and
not in others. What I would like to test is whether these results are due to
the covariance departure from Wishart distribution..
In Bollen book, Structural equations with latent variables, p134-135 :
during estimation, ML need the multi normality in order to deal with Wishart
distribution for the covariance matrix.
My hypothesis (unsure whether this is can be theoretically handled):
- data are not normal, but the covariance show no departure from Wishart,
the ML results are accurate
- data are not normal, but the covariance show significant departure from
Wishart, the ML results are biased..

Best regards & thanks for your help
Naji
-----Message d'origine-----
De : owner-statalist@hsphsun2.harvard.edu
[mailto:owner-statalist@hsphsun2.harvard.edu]De la part de FEIVESON,
ALAN H. (AL) (JSC-SK) (NASA)
Envoyé : jeudi 15 avril 2004 17:56
À : 'statalist@hsphsun2.harvard.edu'
Objet : st: RE: RE: RE: Wishart distribution


Naji - If you have only one such matrix, the answer is that as long as the
matrix is symmetric and positive definite or semi-definite, there is no
test. This is because any symmettirc positive matrix could arise as a sample
of one from some legitimate Wishart distribution.

To see this, suppose you had only one dimension. Then your matrix would be
some number, say 64.5.Your question would be equivalent to asking whether
64.5 could have been generated by sig^2 times a chi-squared random variable
divided by its degress of freedom. The answer, of course is "yes", since
there is always a value of sig^2 that will work.

Al Feiveson

-----Original Message-----
From: owner-statalist@hsphsun2.harvard.edu
[mailto:owner-statalist@hsphsun2.harvard.edu]On Behalf Of Naji Nassar
(MIReS)
Sent: Thursday, April 15, 2004 9:34 AM
To: statalist@hsphsun2.harvard.edu
Subject: st: RE: RE: Wishart distribution


Hi Al,


- How can I test whether an observed covariance (from a sample size) follow
the theoretical distribution?
As input, I have the observed covariance matrix (pxp) and sample size (n)
(not the original data) and the theoretical covariance matrix (pxp).

Best
Naji

-----Message d'origine-----
De : owner-statalist@hsphsun2.harvard.edu
[mailto:owner-statalist@hsphsun2.harvard.edu]De la part de FEIVESON,
ALAN H. (AL) (JSC-SK) (NASA)
Envoyé : jeudi 15 avril 2004 15:08
À : 'statalist@hsphsun2.harvard.edu'
Objet : st: RE: Wishart distribution


Naji - I assume that all you have is S, not the original data - otherwise
you could test if the original data is distributed as multivariate Normal.
Is this the case?
Al Feiveson

-----Original Message-----
From: owner-statalist@hsphsun2.harvard.edu
[mailto:owner-statalist@hsphsun2.harvard.edu]On Behalf Of Naji Nassar
(MIReS)
Sent: Thursday, April 15, 2004 7:34 AM
To: statalist@hsphsun2.harvard.edu
Subject: st: Wishart distribution


Hi all,

Some question about covariance matrix and Wishart dist.
I've a theorical covariance matrix S.
Suppose X RandomNormal(N,p)*CholDecomposition(S) a random correlated
variable.
- What is the theoretical distribution of X covariance (Wishart(S,nobs)?)
- How can I test whether an observed covariance (from a sample size) follow
the theoretical distribution.

Thanks & Best Regards
Naji


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