Matrix F-distribution

Construction of the distribution

• The standard matrix F distribution, with an identity scale matrix ${\displaystyle \mathbf {I} _{p}}$, was originally derived by. When considering independent distributions,

${\displaystyle {\mathbf {\Phi } _{1}}\sim {\mathcal {W}}({\mathbf {I} _{p}},\nu )}$ and ${\displaystyle {\mathbf {\Phi } _{2}}\sim {\mathcal {W}}({\mathbf {I} _{p}},\delta +k-1)}$, and define ${\displaystyle \mathbf {X} ={\mathbf {\Phi } _{2}}^{-1/2}{\mathbf {\Phi } _{1}}{\mathbf {\Phi } _{2}}^{-1/2}}$, then ${\displaystyle \mathbf {X} \sim {\mathcal {F}}({\mathbf {I} _{p}},\nu ,\delta )}$.

• If ${\displaystyle {\mathbf {X} }|\mathbf {\Phi } \sim {\mathcal {W}}^{-1}({\mathbf {\Phi } },\delta +p-1)}$ and ${\displaystyle {\mathbf {\Phi } }\sim {\mathcal {W}}({\mathbf {\Psi } },\nu )}$, then, after integrating out ${\displaystyle \mathbf {\Phi } }$, ${\displaystyle \mathbf {X} }$ has a matrix F-distribution, i.e.,

${\displaystyle f_{\mathbf {X} |\mathbf {\Phi } ,\nu ,\delta }(\mathbf {X} )=\int f_{\mathbf {X} |\mathbf {\Phi } ,\delta +p-1}(\mathbf {X} )f_{\mathbf {\Phi } |\mathbf {\Psi } ,\nu }(\mathbf {\Phi } )d\mathbf {\Phi } .}$
This construction is useful to construct a semi-conjugate prior for a covariance matrix.

• If ${\displaystyle {\mathbf {X} }|\mathbf {\Phi } \sim {\mathcal {W}}({\mathbf {\Phi } },\nu )}$ and ${\displaystyle {\mathbf {\Phi } }\sim {\mathcal {W}}^{-1}({\mathbf {\Psi } },\delta +p-1)}$, then, after integrating out ${\displaystyle \mathbf {\Phi } }$, ${\displaystyle \mathbf {X} }$ has a matrix F-distribution, i.e.,
  ${\displaystyle f_{\mathbf {X} |\mathbf {\Psi } ,\nu ,\delta }(\mathbf {X} )=\int f_{\mathbf {X} |\mathbf {\Phi } ,\nu }(\mathbf {X} )f_{\mathbf {\Phi } |\mathbf {\Psi } ,\delta +p-1}(\mathbf {\Phi } )d\mathbf {\Phi } .}$
  This construction is useful to construct a semi-conjugate prior for a precision matrix.

Marginal distributions from a matrix F distributed matrix

Suppose ${\displaystyle {\mathbf {A} }\sim F({\mathbf {\Psi } },\nu ,\delta )}$ has a matrix F distribution. Partition the matrices ${\displaystyle {\mathbf {A} }}$ and ${\displaystyle {\mathbf {\Psi } }}$ conformably with each other

${\displaystyle {\mathbf {A} }={\begin{bmatrix}\mathbf {A} _{11}&\mathbf {A} _{12}\\\mathbf {A} _{21}&\mathbf {A} _{22}\end{bmatrix}},\;{\mathbf {\Psi } }={\begin{bmatrix}\mathbf {\Psi } _{11}&\mathbf {\Psi } _{12}\\\mathbf {\Psi } _{21}&\mathbf {\Psi } _{22}\end{bmatrix}}}$

where ${\displaystyle {\mathbf {A} _{ij}}}$ and ${\displaystyle {\mathbf {\Psi } _{ij}}}$ are ${\displaystyle p_{i}\times p_{j}}$ matrices, then we have ${\displaystyle {\mathbf {A} _{11}}\sim F({\mathbf {\Psi } _{11}},\nu ,\delta )}$.

Moments

Let ${\displaystyle X\sim F({\mathbf {\Psi } },\nu ,\delta )}$.

The mean is given by: ${\displaystyle E(\mathbf {X} )={\frac {\nu }{\delta -2}}\mathbf {\Psi } .}$

The (co)variance of elements of ${\displaystyle \mathbf {X} }$ are given by:

${\displaystyle \operatorname {cov} (X_{ij},X_{ml})=\Psi _{ij}\Psi _{ml}{\tfrac {2\nu ^{2}+2\nu (\delta -2)}{(\delta -1)(\delta -2)^{2}(\delta -4)}}+(\Psi _{il}\Psi _{jm}+\Psi _{im}\Psi _{jl})\left({\tfrac {2\nu +\nu ^{2}(\delta -2)+\nu (\delta -2)}{(\delta -1)(\delta -2)^{2}(\delta -4)}}+{\tfrac {\nu }{(\delta -2)^{2}}}\right).}$