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I already asked this question on math.stackexchange but got no answer.

For a non-integer number of degrees of freedom $\nu > p-1$, one can simulate the central Wishart distribution $W_p(\nu, \Sigma)$ with the help of the Bartlett decomposition.

How to simulate the noncentral Wishart distribution $W_p(\nu, \Sigma, \Theta)$ for such a $\nu$ (and $\Theta$ a $p\times p$ semi-definite positive matrix)? According to Letac & Massam's tutorial on non central Wishart distributions, this distribution exists for $\Sigma=I_p$, in the sense that there is a distribution whose characteristic function is the same as the one of $W_{p}(``\text{an integer} \geq p``, I_p, \Theta)$ with $\nu$ instead of the integer. For $\Sigma \neq I_p$ I don't know actually.

For $\nu>2p-1$ it suffices to apply the equality $W_p(\nu, \Sigma, \Theta) = W_{p}(\nu-p, \Sigma) \ast W_{p}(p, \Sigma, \Theta)$. How to do in the case when $p-1 < \nu \leq 2p-1$?

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  • $\begingroup$ Namely: math.stackexchange.com/q/2542561 (I believe that in principle the policy, though, is to give your math.SE question a week before asking it here too.) $\endgroup$
    – Jules Lamers
    Commented Nov 30, 2017 at 12:28
  • $\begingroup$ Can I ask for a reference on your statement about the central Wishart distribution? $\endgroup$
    – Alex P. Miller
    Commented Nov 2, 2023 at 16:53

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Efficient Simulation of the Wishart model (2009) shows how to simulate the fractional non-central Wishart distribution for all $\nu>p-1$. See section 6.1.2.b for $\nu\geq p+1$ and page 41 and following for $p-1<\nu<p+1$.

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