Scaled inverse chi-squared distribution

The scaled inverse chi-squared distribution ψ inv- χ 2 ( ν ) {\displaystyle \psi \,{\mbox{inv-}}\chi ^{2}(\nu )} , where ψ {\displaystyle \psi } is the scale parameter, equals the univariate inverse Wishart distribution W − 1 ( ψ , ν ) {\displaystyle {\mathcal {W}}^{-1}(\psi ,\nu )} with degrees of freedom ν {\displaystyle \nu } . This family of scaled inverse chi-squared distributions is linked to the inverse-chi-squared distribution and to the chi-squared distribution: If X ∼ ψ inv- χ 2 ( ν ) {\displaystyle X\sim \psi \,{\mbox{inv-}}\chi ^{2}(\nu )} then X / ψ ∼ inv- χ 2 ( ν ) {\displaystyle X/\psi \sim {\mbox{inv-}}\chi ^{2}(\nu )} as well as ψ / X ∼ χ 2 ( ν ) {\displaystyle \psi /X\sim \chi ^{2}(\nu )} and 1 / X ∼ ψ − 1 χ 2 ( ν ) {\displaystyle 1/X\sim \psi ^{-1}\chi ^{2}(\nu )} .

Source: Wikipedia — Scaled inverse chi-squared distribution (CC BY-SA 4.0)

Scaled inverse chi-squared distribution

The scaled inverse chi-squared distribution ψ inv- χ 2 ( ν ) {\displaystyle \psi \,{\mbox{inv-}}\chi ^{2}(\nu )} , where ψ {\displaystyle \psi } is the scale parameter, equals the univariate inverse Wishart distribution W − 1 ( ψ , ν ) {\displaystyle {\mathcal {W}}^{-1}(\psi ,\nu )} with degrees of freedom ν {\displaystyle \nu } . This family of scaled inverse chi-squared distributions is linked to the inverse-chi-squared distribution and to the chi-squared distribution: If X ∼ ψ inv- χ 2 ( ν ) {\displaystyle X\sim \psi \,{\mbox{inv-}}\chi ^{2}(\nu )} then X / ψ ∼ inv- χ 2 ( ν ) {\displaystyle X/\psi \sim {\mbox{inv-}}\chi ^{2}(\nu )} as well as ψ / X ∼ χ 2 ( ν ) {\displaystyle \psi /X\sim \chi ^{2}(\nu )} and 1 / X ∼ ψ − 1 χ 2 ( ν ) {\displaystyle 1/X\sim \psi ^{-1}\chi ^{2}(\nu )} .

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Source: Wikipedia "Scaled inverse chi-squared distribution" · CC BY-SA 4.0

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