Complex inverse Wishart distribution

If ${\displaystyle \mathbf {S} _{p\times p}}$ is a sample from the complex Wishart distribution ${\displaystyle {\mathcal {CW}}({\mathbf {\Sigma } },\nu ,p)}$ such that, in the simplest case, ${\displaystyle \nu \geq p{\text{ and }}\left|\mathbf {S} \right|>0}$ then ${\displaystyle \mathbf {X} =\mathbf {S} ^{-1}}$ is sampled from the inverse complex Wishart distribution ${\displaystyle {\mathcal {CW}}^{-1}({\mathbf {\Psi } },\nu ,p){\text{ where }}\mathbf {\Psi } =\mathbf {\Sigma } ^{-1}}$.

The density function of ${\displaystyle \mathbf {X} }$ is

${\displaystyle f_{\mathbf {x} }(\mathbf {x} )={\frac {\left|\mathbf {\Psi } \right|^{\nu }}{{\mathcal {C}}\Gamma _{p}(\nu )}}\left|\mathbf {x} \right|^{-(\nu +p)}e^{-\operatorname {tr} (\mathbf {\Psi } \mathbf {x} ^{-1})}}$

where ${\displaystyle {\mathcal {C}}\Gamma _{p}(\nu )}$ is the complex multivariate Gamma function

${\displaystyle {\mathcal {C}}\Gamma _{p}(\nu )=\pi ^{{\tfrac {1}{2}}p(p-1)}\prod _{j=1}^{p}\Gamma (\nu -j+1)}$

The variances and covariances of the elements of the inverse complex Wishart distribution are shown in Shaman's paper above while Maiwald and Kraus[3] determine the 1-st through 4-th moments.

The joint distribution of the real eigenvalues of the inverse complex (and real) Wishart are found in Edelman's paper[4] who refers back to an earlier paper by James.[5] In the non-singular case, the eigenvalues of the inverse Wishart are simply the inverted values for the Wishart. Edelman also characterises the marginal distributions of the smallest and largest eigenvalues of complex and real Wishart matrices.