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.

Shaman finds the first moment to be

${\displaystyle \mathbf {E} [{\mathcal {C}}\mathbf {W^{-1}} ]={\frac {1}{n-p}}\mathbf {\Psi ^{-1}} ,\;n>p}$

and, in the simplest case ${\displaystyle \mathbf {\Psi } =\mathbf {I} _{p\times p}}$, given ${\displaystyle d={\frac {1}{n-p}}}$, then

${\displaystyle \mathbf {\mathbf {E} \left[vec({\mathcal {C}}W_{3}^{-1})\right]} ={\begin{bmatrix}d&0&0&0&d&0&0&0&d\\\end{bmatrix}}}$

The vectorised covariance is

${\displaystyle \mathbf {Cov\left[vec({\mathcal {C}}W_{p}^{-1})\right]} =b\left(\mathbf {I} _{p}\otimes I_{p}\right)+c\,\mathbf {vecI_{p}} \left(\mathbf {vecI_{p}} \right)^{T}+(a-b-c)\mathbf {J} }$

where ${\displaystyle \mathbf {J} }$ is a ${\displaystyle p^{2}\times p^{2}}$ identity matrix with ones in diagonal positions ${\displaystyle 1+(p+1)j,\;j=0,1,\dots p-1}$ and ${\displaystyle a,b,c}$ are real constants such that for ${\displaystyle n>p+1}$

${\displaystyle a={\frac {1}{(n-p)^{2}(n-p-1)}}}$, marginal diagonal variances

${\displaystyle b={\frac {1}{(n-p+1)(n-p)(n-p-1)}}}$, off-diagonal variances.

${\displaystyle c={\frac {1}{(n-p+1)(n-p)^{2}(n-p-1)}}}$, intra-diagonal covariances

For ${\displaystyle \mathbf {\Psi } =\mathbf {I} _{3}}$, we get the sparse matrix:

${\displaystyle \mathbf {Cov\left[vec({\mathcal {C}}W_{3}^{-1})\right]} ={\begin{bmatrix}a&\cdot &\cdot &\cdot &c&\cdot &\cdot &\cdot &c\\\cdot &b&\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot \\\cdot &\cdot &b&\cdot &\cdot &\cdot &\cdot &\cdot &\cdot \\\cdot &\cdot &\cdot &b&\cdot &\cdot &\cdot &\cdot &\cdot \\c&\cdot &\cdot &\cdot &a&\cdot &\cdot &\cdot &c\\\cdot &\cdot &\cdot &\cdot &\cdot &b&\cdot &\cdot &\cdot \\\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &b&\cdot &\cdot \\\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &b&\cdot \\c&\cdot &\cdot &\cdot &c&\cdot &\cdot &\cdot &a\\\end{bmatrix}}}$

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.