Distribution of Trace of non-centered Wishart matrix

Question

I am looking for the distribution of trace of the non-central Wishart matrix with different variations along different axes.

Is there a general formula for such distribution? If not, is there a general formula for the distribution of eigenvalues of such a Wishart matrix?

jvbraun · Accepted Answer · 2014-11-04 18:35:41Z

5

It looks like you can find it in:

It looks like they cover the case of general covariance matrix $\Sigma$ there.

They also give pointers to in the paper to:

A.M. Mathai (1980) Moments of the trace of a noncentral Wishart matrix. Comm. Statist. - Theor. Meth., A9(8), 795--801.

That latter may be useful for computational purposes.

answered Nov 4, 2014 at 18:35

jvbraun

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$\begingroup$ Thank you for your response. I've already encountered these publications and was wondering if there was an expression of PDF that would not rely on the infinite series. Would you have heard anything along those lines? $\endgroup$
– Andrei Kucharavy
Commented Dec 2, 2014 at 15:58
2

$\begingroup$ I have not (yet) seen any expressions of the PDF that do not have infinite series. (Zanella, Chiani, and Win, 2009) gives a closed-form solution for the distribution of ordered eigenvalues. Maybe that can be used to get to the trace? $\endgroup$
– jvbraun
Commented Dec 2, 2014 at 21:44
$\begingroup$ Unfortunately I already saw this paper and wasn't able to extract the information I needed from it. Thank you nonetheless for your pointers. $\endgroup$
– Andrei Kucharavy
Commented Dec 2, 2014 at 21:56
1

$\begingroup$ Looking through some of the literature by A.T. James, T.W. Anderson, and A.G. Constantine, I'm getting the feeling that if there were a nice analytic expression of the distribution of the trace in this setting that it would be visible by now. $\endgroup$
– jvbraun
Commented Dec 3, 2014 at 0:13

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