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ⓘ Complex inverse Wishart distribution. The complex inverse Wishart distribution is a matrix probability distribution defined on complex-valued positive-definite ..




                                     

ⓘ Complex inverse Wishart distribution

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 while the derivation of the inverse is shown by Shaman 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 S p × p = ∑ j = 1 ν G j G j H {\displaystyle \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 {\displaystyle G_{j}} whose Hermitian covariance has complex Wishart distribution S ∼ C W Σ, ν, p {\displaystyle \mathbf {S} \sim {\mathcal {CW}}\mathbf {\Sigma },\nu,p} with mean value Σ and ν {\displaystyle \mathbf {\Sigma } {\text{ and }}\nu } degrees of freedom, then the pdf of X = S − 1 {\displaystyle \mathbf {X} =\mathbf {S^{-1}} } follows the complex inverse Wishart distribution.

                                     

1. Density

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

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

f x = | Ψ | ν C Γ p ν | x | − ν + p e − tr ⁡ Ψ x − 1 {\displaystyle f_{\mathbf {x} }\mathbf {x}={\frac {\left|\mathbf {\Psi } \right|^{\nu }}pp-1}\prod _{j=1}^{p}\Gamma \nu -j+1}
                                     

2. Moments

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

                                     

3. Eigenvalue distributions

The joint distribution of the real eigenvalues of the inverse complex and real Wishart are found in Edelmans paper who refers back to an earlier paper by James. 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.

                                     
  • 1365 - 2311.1927.tb00074.x. hdl: 2440 15100. with J Wishart Fisher, R. A. Wishart J. 1927 On the Distribution of the Error of an Interpolated Value, and
  • the Wishart distribution which is the conjugate prior of the precision matrix inverse covariance matrix for a multivariate Gaussian distribution Mult
  • in: Bob Rowthorn, Capitalism, Conflict and Inflation. London: Lawrence Wishart 1980. See also: Howard King, A History of Marxian Economics, Vol. 2
  • biological, geological, and chemical processes in the environment using the distribution and relative abundance of hydrogen isotopes. There are two stable isotopes

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