Complex inverse Wishart distribution information

Complex inverse Wishart Distribution
Notation
Parameters	degrees of freedom (real); , scale matrix (pos. def.)
Support	is p × p positive definite Hermitian
PDF	is the complex multivariate gamma function; is the trace function;
Mean	for
Variance	see below

The complex inverse Wishart distribution is a matrix probability distribution defined on complex-valued positive-definite matrices and is the complex analog of the real inverse Wishart distribution. The complex Wishart distribution was extensively investigated by Goodman^[1] while the derivation of the inverse is shown by Shaman^[2] and others. It has greatest application in least squares optimization theory applied to complex valued data samples in digital radio communications systems, often related to Fourier Domain complex filtering.

Letting $\mathbf {S} _{p\times p}=\sum _{j=1}^{\nu }G_{j}G_{j}^{H}$ be the sample covariance of independent complex p-vectors $G_{j}$ whose Hermitian covariance has complex Wishart distribution $\mathbf {S} \sim {\mathcal {CW}}(\mathbf {\Sigma } ,\nu ,p)$ with mean value $\mathbf {\Sigma } {\text{ and }}\nu$ degrees of freedom, then the pdf of $\mathbf {X} =\mathbf {S^{-1}}$ follows the complex inverse Wishart distribution.

^ Goodman, N R (1963). "Statistical Analysis Based on a Certain Multivariate Complex Gaussian Distribution: an Introduction". Ann. Math. Statist. 34 (1): 152–177. doi:10.1214/aoms/1177704250.
^ Shaman, Paul (1980). "The Inverted Complex Wishart Distribution and its Application to Spectral Estimation". Journal of Multivariate Analysis. 10: 51–59. doi:10.1016/0047-259X(80)90081-0.

[1] Goodman, N R (1963). "Statistical Analysis Based on a Certain Multivariate Complex Gaussian Distribution: an Introduction". Ann. Math. Statist. 34 (1): 152–177. doi:10.1214/aoms/1177704250.

[2] Shaman, Paul (1980). "The Inverted Complex Wishart Distribution and its Application to Spectral Estimation". Journal of Multivariate Analysis. 10: 51–59. doi:10.1016/0047-259X(80)90081-0.

Complex inverse Wishart distribution information

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Notation	${\mathcal {CW}}^{-1}({\mathbf {\Psi } },\nu ,p)$
Parameters	$\nu >p-1$ degrees of freedom (real) $\mathbf {\Psi } >0$ , $p\times p$ scale matrix (pos. def.)
Support	$\mathbf {X}$ is p × p positive definite Hermitian
PDF	${\frac {\left\|\mathbf {\Psi } \right\|^{\nu }}{{\mathcal {C}}\Gamma _{p}(\nu )}}\left\|\mathbf {x} \right\|^{-(\nu +p)}e^{-\operatorname {tr} (\mathbf {\Psi } \mathbf {x} ^{-1})}$ ${\mathcal {C}}\Gamma _{p}$ is the complex multivariate gamma function ${\mathcal {C}}\Gamma _{p}(\nu )=\pi ^{{\tfrac {1}{2}}p(p-1)}\prod _{j=1}^{p}\Gamma (\nu -j+1)$ $\operatorname {tr}$ is the trace function
Mean	${\frac {\mathbf {\Psi } }{\nu -p}}$ for $\nu >p+1$
Variance	see below