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The non-central chi-square distribution with $\nu$ degrees of freedom has a density which can be expressed as $$ f(x) = \sum_{i=0}^{\infty} c_i f_{\nu + 2i}(x), $$ where $c_i$ is a function of the non-centrality parameter and $i$, and where $f_{j}(x)$ is the density of a chi-square distribution with $j$ degrees of freedom. (The non-central chi-squared distribution is a Poisson weighted mixture of central chi-squared distributions).

My question is whether this property extends to the multivariate case, the non-central Wishart distribution: can we express the non-central Wishart as a weighted mixture of central Wishart distributions?

(My guess is not, except in the scalar case; if it does hold, however, and the $c_i$ are easy to compute, I would like to use this property to compute the moments of the non-central Wishart.)

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  • $\begingroup$ It looks like Theorem 1 of Li and Geng provides something like what I would need, for the case where the non-centrality matrix is rank 1. $\endgroup$
    – Steven Pav
    Commented Mar 23, 2018 at 4:53

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arXiv:1512.08159 shows that the distribution of the scalar Schur complement in a noncentral Wishart matrix is a mixture of central chi-square distributions with different degrees of freedom. For the case of a rank-1 noncentrality matrix, the weights of the mixture representation arise from a noncentral beta mixture of Poisson distributions.

If the goal is to calculate the moments of the noncentral Wishart distribution, the graph representation of arXiv:0912.0577 seems convenient. (See also the tutorial On the moment formulas for the noncentral Wishart distributions.)

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