# Distribution of Trace of non-centered Wishart matrix

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?

It looks like you can find it in:

S. Kourouklis and P.G. Moschopoulos (1985) On the distribution of the trace of a non-central Wishart. Metron XLIII(1--2): 85--92.

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

They also give pointers to in the paper to:

Mathai, A.M. and Pillai, K.C.S. (1982) Further results on the trace of a non-central Wishart matrix, Comm. Statist.-Theor. Meth., A 11, 1077-1086.

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.

• 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? – Andrei Kucharavy Dec 2 '14 at 15:58
• 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? – jvbraun Dec 2 '14 at 21:44
• Unfortunately I already saw this paper and wasn't able to extract the information I needed from it. Thank you nonetheless for your pointers. – Andrei Kucharavy Dec 2 '14 at 21:56
• 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. – jvbraun Dec 3 '14 at 0:13