I'm currently dealing with a Gibbs sampler of the multivariate generalized inverse Gaussian distribution (MGIG). In order to check the correctness of the sampler, I'd like to know the expected value of this distribution. I read a couple of papers involving the MGIG but none provides the expectation.
Or, better, I'd like to know what is the characteristic function of the MGIG, or the moments generating function.
I have a partial answer.
Let $z \in \mathbb{R}^d$, let $A$ a positive definite $d\times d$ matrix, let $b > 0$, and let $q > (d-1)/2$. Define $p = q + (1-d)/2$ and $H = zz^{t}$. The following is claimed in the paper "A Bayesian approach for clustering skewed data using mixtures of multivariate normal-inverse Gaussian distributions" by Yang & al.
Let $x \sim GIG(-p, z^tAz, b)$ (generalized inverse Gaussian) and $W \sim \mathcal{W}_d(q,A)$ (Wishart) be independant. Then ${(xH + W)}^{-1}$ has the $MGIG(-q, bH, A)$ distribution.
So one has the expectation of the inverse of $MGIG(-q, bH, A)$. This is $\mathbb{E(x)}H + \mathbb{E}(W)$, and these two expectations are known.
Now I believe I saw somewhere that this inverse is also $MGIG$. I'll edit this answer if I retrieve the statement of this fact. Edit: indeed; it is $MGIG(q, A, bH)$.
This is a very partial answer because $bH$ is of rank one and $-q < (1-d)/2 \leqslant 0$.
