Matrix F-distribution

In statistics, the matrix F distribution (or matrix variate F distribution) is a matrix variate generalization of the F distribution which is defined on real-valued positive-definite matrices. In Bayesian statistics it can be used as the semi conjugate prior for the covariance matrix or precision matrix of multivariate normal distributions, and related distributions.

Density

The probability density function of the matrix $F$ distribution is:

$f_{𝐗}(𝐗;\mathrm{\Psi },\nu ,\delta )=\frac{{\mathrm{\Gamma }}_p\left(\frac{\nu +\delta +p-1}{2}\right)}{{\mathrm{\Gamma }}_p\left(\frac{\nu }{2}\right){\mathrm{\Gamma }}_p\left(\frac{\delta +p-1}{2}\right)|\mathrm{\Psi }{|}^{\frac{\nu }{2}}}|𝐗{|}^{\frac{\nu -p-1}{2}}|{\mathbf{\text{I}}}_p+𝐗{\mathrm{\Psi }}^{-1}{|}^{-\frac{\nu +\delta +p-1}{2}}$

where $𝐗$ and $\mathrm{\Psi }$ are $p\times p$ positive definite matrices, $|\cdot |$ is the determinant, Γ_p(⋅) is the multivariate gamma function, and ${\mathbf{\text{I}}}_p$ is the p × p identity matrix.

Properties

Construction of the distribution

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

${\mathrm{\Phi }}_1\sim 𝒲({𝐈}_p,\nu )$ and ${\mathrm{\Phi }}_2\sim 𝒲({𝐈}_p,\delta +k-1)$, and define $𝐗={{\mathrm{\Phi }}_2}^{-1/2}{\mathrm{\Phi }}_1{{\mathrm{\Phi }}_2}^{-1/2}$, then $𝐗\sim ℱ({𝐈}_p,\nu ,\delta )$.

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

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

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

Marginal distributions from a matrix F distributed matrix

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

$𝐀=\left[\begin{array}{cc}{𝐀}_{11}& {𝐀}_{12}\\ {𝐀}_{21}& {𝐀}_{22}\end{array}\right],\mathrm{\Psi }=\left[\begin{array}{cc}{\mathrm{\Psi }}_{11}& {\mathrm{\Psi }}_{12}\\ {\mathrm{\Psi }}_{21}& {\mathrm{\Psi }}_{22}\end{array}\right]$

where ${𝐀}_{ij}$ and ${\mathrm{\Psi }}_{ij}$ are $p_i\times p_j$ matrices, then we have ${𝐀}_{11}\sim F({\mathrm{\Psi }}_{11},\nu ,\delta )$.

Moments

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

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

The (co)variance of elements of $𝐗$ are given by:

$\mathrm{cov}(X_{ij},X_{ml})={\mathrm{\Psi }}_{ij}{\mathrm{\Psi }}_{ml}\frac{2{\nu }^2+2\nu (\delta -2)}{(\delta -1)(\delta -2{)}^2(\delta -4)}+({\mathrm{\Psi }}_{il}{\mathrm{\Psi }}_{jm}+{\mathrm{\Psi }}_{im}{\mathrm{\Psi }}_{jl})(\frac{2\nu +{\nu }^2(\delta -2)+\nu (\delta -2)}{(\delta -1)(\delta -2{)}^2(\delta -4)}+\frac{\nu }{(\delta -2{)}^2}).$

Related distributions

• The matrix F-distribution has also been termed the multivariate beta II distribution. See also, for a univariate version.
• A univariate version of the matrix F distribution is the F-distribution. With $p=1$ (i.e. univariate) and $\mathrm{\Psi }=1$, and $x=𝐗$, the probability density function of the matrix F distribution becomes the univariate (unscaled) F distribution:
  $f_{x\mid \nu ,\delta }(x)=\mathrm{B}{(\frac{\nu }{2},\frac{\delta }{2})}^{-1}{\left(\frac{\nu }{\delta }\right)}^{\nu /2}{x}^{\nu /2-1}{(1+\frac{\nu }{\delta }x)}^{-(\nu +\delta )/2},$

• In the univariate case, with $p=1$ and $x=𝐗$, and when setting $\nu =1$, then $\sqrt{x}$ follows a half t distribution with scale parameter $\sqrt{\psi }$ and degrees of freedom $\delta$. The half t distribution is a common prior for standard deviations

See also

• Inverse matrix gamma distribution
• Matrix normal distribution
• Wishart distribution
• Inverse Wishart distribution
• Complex inverse Wishart distribution